2) Drs. Mustafa, M.Pd., selaku Kepala Dinas Pendidikan Kota Langsa

Ucapan Terima Kasih Syukur Alhamdulillah, akhirnya kami dapat menyelesaikan Lembar Kerja Siswa (LKS) Matematika untuk SMP/MTs Kelas VIII Semester 1 dengan bantuan berbagai pihak. Untuk itu, pada kesempatan ini kami ingin mengucapkan terima kasih yang sebesar-besarnya kepada: 1)

Direktorat Pembinaan Pendidik dan Tenaga Kependidikan Pendidikan Dasar, Dirjen. Pendidikan Dasar, Kementerian Pendidikan Dan Kebudayaan, yang telah memberikan bantuan dana Pengembangan karir PTK Dikdas: MGMP SMP tahun 2012 guna terselenggaranya penyusunan LKS Matematika ini.

2) Drs. Mustafa, M.Pd., selaku Kepala Dinas Pendidikan Kota Langsa. 3) Hardani, S.Pd., selaku Ketua MGMP Matematika tingkat SMP Wilayah Timur Kota Langsa,

4) Intan Yuliani, S.Pd., selaku Sekretaris MGMP Matematika tingkat SMP Wilayah Timur Kota Langsa, 5) Muhammad Yusuf, S.Pd., selaku bendahara MGMP Matematika tingkat SMP Wilayah Timur Kota Langsa, 6) Yenny Suzana, M.Pd., selaku pembimbing dalam penyusunan dan penyelesaian LKS Matematika ini, 7) Segenap peserta sebagai Tim Penyusun LKS pada Workshop Pengembangan Karir Pendidik dan Tenaga Kependidikan (PTK) Pendidikan Dasar (Dikdas) MGMP Matematika Matriks tingkat SMP Wilayah Timur Kota Langsa tahun 2012, dan 8) Semua pihak yang telah membantu dalam penyelesaian Lembar Kerja Siswa (LKS) Matematika ini yang tidak dapat disebutkan satu persatu. Semoga LKS Matematika ini dapat memberi manfaat bagi siswa, guru, dosen, dan praktisi di bidang pendidikan, serta bermanfaat bagi masyarakat luas pada umumnya. Atas bantuan yang telah Bapak/Ibu berikan mendapat balasan yang setimpal dari Allah S.W.T. Amiin.

Langsa, Juli 2012

Tim Penyusun

KATA PENGANTAR

Puji syukur kami panjatkan kehadirat Allah SWT karena berkat rahmat dan hidayahNya, sehingga kami dapat menyelesaikan penulisan Lembar kerja Siswa (LKS) Matematika untuk SMP/MTs Kelas VIII Semester 1 yang merupakan salah satu produk hasil Workshop Pengembangan Karir Pendidik dan Tenaga Kependidikan (PTK) Pendidikan Dasar (Dikdas) MGMP Matematika Matriks tingkat SMP Wilayah Timur Kota Langsa tahun 2012. LKS Matematika ini merupakan wujud kerja Guru dalam mengembangkan karir guna menjadi Guru yang profesional. LKS Matematika ini disusun untuk menuntun siswa agar menemukan sendiri suatu konsep dalam matematika, dengan cara yang lebih mudah dipahami dan kontekstual. Dengan adanya LKS Matematika ini diharapkan siswa dapat berperan aktif dalam proses pembelajaran. Secara keseluruhan, LKS Matematika ini terdiri dari 5 BAB, dan masing-masing BAB terbagi menjadi beberapa kegiatan Pembelajaran. Kami menyadari bahwa dalam penyusunan LKS Matematika ini masih jauh dari kesempurnaan. Untuk itu, kritik dan saran demi perbaikan lebih lanjut sangat kami harapkan. Mudah-mudahan LKS matematika ini dapat memberi manfaat bagi siswa, guru, dan praktisi pendidikan.

Langsa, Juli 2012

Tim Penyusun

KATA PENGANTAR

i

DAFTAR ISI

o o o o o o o

Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran

1.1 ........................................ 1 1.2 ...................................... 4 1.3 ...................................... 6 1.4 .................................... 10 1.5 .................................... 12 1.6 .................................... 15 1.7 .................................... 18

o o o o o o

Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran

2.1 ................................... 20 2.2 ................................... 23 2.3 ................................... 26 2.4 ................................... 28 2.5 .................................... 31 2.6 ................................... 33

o o o o o

Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran

3.1 ................................... 39 3.2 ................................... 46 3.3 .................................... 51 3.4 ................................... 55 3.5 ................................... 58

o o o o o

Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran

4.1 .................................... 61 4.2 ................................... 66 4.3 ................................... 68 4.4 ................................... 72 4.5 ................................... 74

o o o o o o o

Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran Pembelajaran

5.1 ................................... 77 5.2 ................................... 82 5.3 ................................... 85 5.4 ................................... 90 5.5 ................................... 93 5.6 ................................... 95 5.7 ................................... 97

DAFTAR ISI

ii

BAB 1 FAKTORISASI BENTUK ALJABAR Pembelajaran 1.1 Standar Kompetensi

= Memahami bentuk aljabar, relasi, fungsi dan persamaan garis lurus.

Kompetensi Dasar

= Melakukan operasi aljabar.

Indikator

= 1. Menjelaskan pengertian koefisien, variabel, konstanta, suku satu, suku dua dan suku tiga dalam variabel sama atau berbeda. 2. Menyederhanakan bentuk aljabar suku satu, suku dua dan suku banyak.

Tujuan Pembelajaran = Siswa dapat menyederhanakan bentuk aljabar suku satu, suku dua dan suku banyak.

Bentuk Aljabar Pada buku kelas VII telah dibahas tentang pengertian aljabar, koefisien, konstanta, variabel, suku dan faktor.

2a + 5 merupakan bentuk aljabar. Dari bentuk aljabar tersebut, 2 disebut ................................... a disebut ................................... 5 disebut ................................... Perhatikan bentuk-bentuk aljabar berikut: 1) 3a disebut bentuk aljabar suku satu (suku tunggal) 2) 3k + 5 disebut bentuk aljabar suku dua (binom), yaitu: suku pertama 3k dan suku kedua 5 3)

6x2 + 4xy - y disebut bentuk aljabar suku .............................., yaitu: suku pertama............., suku kedua .............. dan suku ketiga ...................

4)

7a2b - 6a2 - 5a + 3b disebut bentuk aljabar suku .............., yaitu: ........, ......., ....... dan .......

Jadi, suku merupakan kumpulan bilangan-bilangan yang dipisahkan oleh ........................................

Operasi Penjumlahan dan Pengurangan Perhatikan uraian berikut ini. Mutia memiliki 9 buku tulis dan 3 buku gambar. Jika buku tulis dinyatakan dengan x dan buku gambar dinyatakan dengan y maka banyaknya buku mutia adalah 9x+3y. Selanjutnya, jika Mutia diberi kakaknya 2 buku tulis dan 4 buku gambar maka banyaknya buku mutia sekarang adalah: 11x + 7y 

Hasil ini diperoleh dari (9x + 3y) + (2x + 4y). 9x + 3y dan 2x + 4y merupakan bentuk aljabar.

Pada bentuk aljabar, suku-suku yang dapat dijumlahkan dan dikurangkan hanyalah suku-suku sejenis saja. Suku-suku sejenis adalah suku-suku dengan variabel dan pangkat variabel yang sama.

BAB 1 FAKTORISASI BENTUK ALJABAR

1

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh

Langkah-langkah untuk menyederhanakan bentuk aljabar suku satu, suku dua, dan suku banyak yaitu: 1) Kelompokkan suku-suku sejenis 2) Jumlahkan atau kurangkan koefisien suku-suku yang sejenis tersebut.

Sederhanakanlah bentuk aljabar berikut ini ! 1)

2x + 3y + 3x – y

Penyelesaian : Kelompokkan suku-suku sejenis 2x + 3y + 3x – y = 2x + ......... + ......... – y =

(......... + 3) x + (3 – 1) .........

Jumlahkan atau kurangkan koefisien suku-suku yang sejenis tersebut, menjadi: 2x + 3y + 3x – y 2)

=

6a2 - 2a2 + 2a - 7a = =

5x + ...... y (6 – 2) ....... + ( ........ – 7) a 4a2 – 5 ........

Selain dengan cara di atas, penjumlahan dan pengurangan pada bentuk aljabar dapat dihitung dengan metode bersusun ke bawah. 3)

Jumlahkan 4x2 – 5x + 4 dan 3x2 + 2x – 6, dengan metode bersusun : 4x2 – 5x + 4 3x2 + 2x – 6 .............................

4)

Kurangkan 2p – 5 dari 10p + 11

Penyelesaian: 10p + 11 – (2p - 5)

= 10p + 11 – 2p + 5 = ........ – 2p + 11 +.......... = .......... + ..........

1)

Tentukan variabel dan koefisien dari masing-masing variabel dan banyak suku bentuk aljabar berikut: a) 3a – 7b c) 3p2 – 5pq + 3q2 .................................................................................................

................................................................................................................................................

.................................................................................................

................................................................................................................................................

.................................................................................................

................................................................................................................................................

.................................................................................................

................................................................................................................................................

b)

2

2x2y + 5xy2

d)

5y (y2 + 3) - 7

.................................................................................................

................................................................................................................................................

.................................................................................................

...............................................................................................................

.................................................................................................

...............................................................................................................

.................................................................................................

................................................................................................................................................

LKS MATEMATIKA KELAS VIII SEMESTER 1

“Memahami lebih baik daripada sekadar membaca”

2)

Sederhanakan bentuk aljabar berikut: a) 10a – 7b + 3a + 2b

b) 2p2 – 5q + 4p2 – 5q

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

a).

Jumlahkanlah bentuk aljabar 3x2 + 7xy - y dan 3x2 - 2xy + 5y

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b)

Jumlahkan dengan metode bersusun bentuk aljabar 4x + 2y – 3z dan 2x – 7y – 6z

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

c)

Kurangkanlah: (8m + 4) dari (9m + 12)

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

d)

Kurangkanlah -2y2 + 4y + 5 dari 10y2 – 12y + 7

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Arman mempunyai 5 buah robot dan 8 buah mobil-mobilan. Jika Arman diberi 2 buah robot oleh ibu dan 3 mobil-mobilannya ia berikan kepada Arif, berapa sisa robot dan mobil Arman! Nyatakan dalam bentuk aljabar. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

5)

Bu Winda membeli 4 kg tepung, 3 kg wortel dan 6 kg tomat. Karena terlalu lama disimpan 2kg tepung, 1 kg wortel dan 2 kg tomat ternyata busuk. Tentukan tepung, wortel, dan tomat yang tersisa! Nyatakan dalam bentuk aljabar. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


3


Pembelajaran 1.2 Standar Kompetensi


Kompetensi Dasar


Indikator

= Menyelesaikan operasi perkalian dan pembagian dari suku satu dan suku dua

Tujuan Pembelajaran = Siswa dapat menyelesaikan operasi perkalian dan pembagian pada bentuk aljabar

Perkalian suku satu dengan suku dua Perkalian suku satu dengan suku dua dapat dilakukan dengan menggunakan sifat distributif berikut: a(x + y) = ax + ay a(x – y) = ax – ay

Tentukanlah hasil perkalian 4 (2a + 3)

Penyelesaian : 4 (2a + 3)

=

(4 x ..........) + (4 x.............)

=

.......... + ..........

Perkalian suku dua dengan suku dua Perkalian suku dua dengan suku dua dapat diselesaikan dengan menggunakan 3 cara yaitu:

o Cara 1 : Menggunakan kartu Adapun langkah – langkah kegiatan perkalian suku dua dengan suku dua dengan menggunakan kartu adalah sebagai berikut: 1) Persiapkan dua jenis kartu dengan warna yang berbeda, misalkan kartu berwarna putih dan biru. 2) Kemudian guntinglah kartu berwarna tersebut dengan ukuran 6 x 6, 6 x 3 dan 3 x 3 sebanyak 20 lembar untuk masing – masing kartu. 3) Untuk kartu putih : Kartu dengan ukuran 6 x 6 dimisalkan dengan x2. Kartu dengan ukuran 6 x 3 dimisalkan dengan x. Kartu dengan ukuran 3 x 3 dimisalkan dengan 1. 4) Untuk kartu Biru : Kartu dengan ukuran 6 x 6 dimisalkan dengan -x2. Kartu dengan ukuran 6 x 3 dimisalkan dengan –x. Kartu dengan ukuran 3 x 3 dimisalkan dengan -1.

dan x2

x

1

Selesaikanlah : ( x + 3) ( x - 2 )

4


-x2

-x

-1


Penyelesaian : (x + 3) (x - 2) x

-2

x

3

= x2 + 3x – ............ – ........

Jadi, ( x + 3) ( x - 2)

= ............ + x - .........

o Cara 2: Menggunakan Sifat Distribusi Selesaikanlah (x + 3) (x - 2)

Penyelesaian : ( x + 3) ( x - 2)

=

x (x - 2) + ....(x - 2)

=

x2 + .... + .... - 6

=

x2 + .... - ....

Secara umum perkalian bentuk aljabar suku dua dengan suku dua dapat ditulis dengan menggunakan skema :

= a(..... + .....) + b(..... + .....) = ac + ad + ..... + .....

Pembagian Bentuk Aljabar Pembagian dari dua atau lebih bentuk aljabar dalam bentuk yang sederhana adalah jika bentukbentuk aljabar tersebut memiliki faktor-faktor yang sama.

1)

1)

2)

Sederhanakanlah bentuk perkalian suku satu dengan suku dua pada bentuk aljabar berikut ini: a) 3 (a + 2) b) 2x (x – 5) ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


5


2)

Sederhanakanlah bentuk perkalian suku dua dengan suku dua berikut dengan 3 cara yaitu: cara skema, tabel dan kartu. a) ( x + 1 ) ( x + 4 ) ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b)

(x–3)(x–2)

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

c)

( 2x + 3 ) ( x – 1 )

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................. ... ............................................................................................................................................................................................................................... ..

3)

Sederhanakanlah: a) 10ab : 2b ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................. ... .............................................................................................................................................................................................................................. ... .............................................................................................................................................................................................................................. ... ............................................................................................................................................................................................................................... .. b)

64x2y2 : 4xy

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................. ... .............................................................................................................................................................................................................................. ... .............................................................................................................................................................................................................................. ... .............................................................................................................................................................................................................................. ... ............................................................................................................................................................................................................................... ..

Nilai Kognitif

6

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= Menyelesaikan operasi pangkat dari suku satu dan suku dua

Tujuan Pembelajaran = Siswa dapat menyelesaikan operasi pangkat dari suku satu dan suku dua

Perpangkatan suku satu Pangkat dari suatu bentuk aljabar adalah perkalian bentuk aljabar dengan dirinya sendiri, sebanyak pangkat yang tertera pada bentuk aljabar tersebut. Dengan kata lain pangkat merupakan perkalian berulang.

22

= 2x2

3

a

= ............ x ............ x ............

(3a)2

= (3a) x ............... 2

(3a + 5)

(a + b)2 3

(p + q)

= (3a + 5) (........... + ........) = (.......... + ...........) (.......... + .........) = (........................) (........................) (........................)

Bentuk aljabar di atas disebut perpangkatan suku dua. Tuliskan contoh lain dari perpangkatan suku dua: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Perpangkatan suku dua Dari contoh di atas, tentukanlah: (a + b)2

= (a + b) (a + b) = a2 + ab + ab + b2 = a2 + ............ + ............

(a + b)3

= (a + b) (a + b)2 = (a + b) (a2 + 2ab + b2) = a3 + ............ a2b + ab2 + a2b + ............ + b3 = a3 + ............ + ............ + ............

Jika dilihat dari kegiatan diatas dan seterusnya maka akan diperoleh pola dari koefisien-koefisien (a + b)n yang disebut koefisien binomial. Koefisien dari perpangkatan suku dua seperti pada contoh di atas dapat direpresentasikan dalam bilangan segitiga pascal, yaitu sebagai berikut:


7


1 1 1

     

1 2

1

1 3 3 1 1 4 6 4 1 ...................................................................

(a + (a + (a + (a + (a + dst

b)0 = b)1 = b)2 = b)3 = b)4 =

1 a+b 1a2 + 2ab + 1b2 1a3 + 3a2b + 3ab2 + 1b3 1a4 + 4a3b + 6a2b2 + 4ab3 + 1b4

Selesaikanlah perpangkatan suku dua berikut ini: 1)

(2a + 4)2

= (2a)2 + 2(..............)(..............) + (..............)2 = 4a2 + .............. + 16

2)

(p – 2q)3

= 1p3 + 3(p)2 (..............) + 3p (..............)2 + 1(-2q)3 = p3 + (-6p2q) +.............. + (-8q3) = .............. - 6p2q + 12pq2 - ..............

Latihan 1) Tentukan hasil pemangkatan bentuk aljabar berikut: a) (4x)2

b) (-5p2q)3

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2) Tentukan hasil dari: a) (2q+3)2

b) (a-2b)2

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b)

(x+2)3

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

8



3) Bu Asni mempunyai kebun berbentuk persegi, dengan panjang sisinya (X+5). a) Nyatakan luas kebun Bu Asni! ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b) Apakah luas kebun Bu Asni merupakan bentuk perpangkatan? .................................................................................................................................................................................................................................

c) Jika merupakan bentuk perpangkatan, perpangkatan suku berapakah luas kebun Bu Asni? .................................................................................................................................................................................................................................

d) Nyatakan luas kebun Bu Asni dengan menggunakan operasi penjumlahan dan pengurangan. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


9




Kompetensi Dasar


Indikator

= a). Memfaktorkan suku bentuk aljabar sampai dengan suku tiga b). Pemfaktoran selisih dua kuadrat

Tujuan Pembelajaran = Siswa dapat memfaktorkan

Faktor-faktor Suku Aljabar Berapakah faktor persekutuan dari 6 dan 8?

penyelesaian: Faktor- faktor 6 : 1, 2, ........, ......... Faktor- faktor 8 : 1, 2, ........, ......... Faktor persekutuan dari 6 dan 8 adalah 1 dan 2. Oleh karena itu 1 < 2 maka 2 dikatakan sebagai faktor persekutuan terbesar (FPB) dari 6 dan 8.

Faktorisasi Bentuk ax ± b Cara untuk memfaktorkan bentuk aljabar ax ± b adalah sebagai berikut : 1) Carilah faktor persekutuan setiap suku 2) Bagilah bentuk aljabar tersebut dengan faktor persekutuan terbesar dari setiap sukunya.

 Faktorkanlah bentuk aljabar 6b + 8

Penyelesaian : carilah faktor persekutuan dari 6b dan 8, kamu telah mengetahui bahwa FPB dari 6 dan 8 adalah 2, kemudian bagilah setiap suku dengan FPB tersebut: dan Dengan demikian, pemfaktoran dari 6b+8 adalah 2(3b + 4) atau 6b+8=2(3b + 4).

Faktorisasi bentuk selisih dua kuadrat Bentuk x2 – y2 dinamakan bentuk selisih dua kuadrat. Faktorisasi bentuk x 2 – y2 adalah sebagai berikut: x2 – y 2 = ( x + y ) ( x – y ) untuk membuktikan persamaan diatas, coba kamu perhatikan uraian berikut : (x+y)(x–y)

= ( x + y ) x + ( x + y ) ( -y ) = x2 + ........... – xy – ........... = x2 – y2

10



Faktorkanlah: 4p2 – 25

Penyelesaian: 4p2 – 25

1)

=

(2p) 2 – (5)2

=

(2p + .........) (......... – 5)

Faktorkanlah bentuk-bentuk aljabar suku dua berikut! a) 8a – 2 b) x2 + 10x ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Faktorkanlah bentuk-bentuk aljabar suku tiga berikut! a) 10 m2 + 15 mn – 35 m b) x2 + 10x + 25 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Faktorkan selisih dan kuadrat berikut! b) 25x2 – 16y2

b) 16c2 – 9a2

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

c)

4x2 – 9y2

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


11




Kompetensi Dasar


Indikator

= a). Pemfaktoran bentuk kuadrat b). Pemfaktoran bentuk ax + bx + c jika a≠1

Tujuan Pembelajaran = Siswa dapat memfaktorkan

Pemfaktoran bentuk kuadrat o

Pemfaktoran bentuk x2 + 2xy + y2 dan x2 - 2xy + y2

Pada bab lalu kamu sudah mempelajari perkalian dua suku seperti; (a + b) (a + b) = a(a + b) + b(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 Sekarang jika dibalik, didapat: a2 + 2ab + b2 = (a + b)(a + b) a2 + 2ab + b2 = (a + b)2 Bentuk diatas disebut bentuk .......................................... sempurna.

o Pemfaktoran tiga suku

1)

faktorkan a2+ 10a +25

Penyelesaian: a2+ 10a +25

= (a)2+ 2a(5) + 52 = (a + 5)2

(x + y)2

= (x + y) (x + y) = x (x + y) + y (x + y) = x2 + ..............+ xy + ............... = ......... + 2xy + y2

Perkalian yang diuraikan diatas disebut pengkuadratan suku dua dan hasilnya x 2 + 2xy + y2 disebut suku tiga bentuk kuadrat sempurna, yang mana bila difaktorkan dan disederhanakan maka kembali kebentuk semula, yakni suku dua yang dikuadratkan (x + y)2 2)

Faktorkanlah x2 + 6x + 9

Penyelesaian: karena (

x 6)

x2 + 6x + 9

= 32

, maka

= x2 + 6x +32 = (x + 3) (x + 3) = (x + ............)2

12


atau


menggunakan hukum distributif, diperoleh: x2 + 6x + 9

= x2 + 2(3x) + 32 = ..............+ ............ + 3x + ............. = x (x +...........) +3 (x +...........) = (x + ...........) (x + ...........) = (x + 3)2

Pemfaktoran bentuk ax2 + bx + c o Memfaktorkan bentuk ax2 + bx + c jika a = 1 Faktorisasi bentuk ax2 + bx + c adalah (x + p) (x + q) dengan b = p + q dan c = p x q Coba kamu perhatikan bentuk aljabar berikut: (x + 2) (x + 5)

= (x + 2) x + (x +2) 5 = x2 + .........+ ..........+ 10 = .......... + 7x + 10

Koefisien suku kedua pada bentuk aljabar diatas yaitu 7 merupakan hasil penjumlahan dua konstanta, yaitu: 2 dan 5. Adapun suku ketiga yaitu: 10 merupakan hasil perkalian dua konstanta yaitu: 10 = 2 x 5

 Faktorkanlah bentuk x2 + 8x + 15

Penyelesaian:

x2 + 8x + 15

= x2 + (p + q)x + pq = (x + p) (x + q)

sehingga diperoleh nilai p = 3 dan q = 5 (karena 3 + ........ = 8 dan 3x5 = .............)

Tentukan nilai p dan q terlebih dahulu: p+q =8 p x q = 15

Jadi, x2 + 8x + 15 = (x + ............) (.......... + 5)

o Memfaktorkan bentuk ax2 + bx + c, jika a≠1 Setelah kamu mempelajari pemfaktoran bentuk ax2 + bx + c untuk a = 1, sekarang muncul pertanyaan bagaimana memfaktorkan bentuk ax2 + bx + c, jika a≠1. Untuk menjawab pertanyaan tersebut, amatilah contoh soal dibawah ini.

 Faktorkanlah 3x2 + 13x + 10 = 0

Penyelesaian: 3x2 + 13x + 10 = 0 a = 3,

b = 13,

dengan menggunakan hukum distributif, diperoleh: c = 10

3x2 + 13x + 10 = 3x2 + 3x +10x +10

p + q = 13

= 3x (....... + 1) + ........ (x + ........)

p x q = a x c = 3 x 10

= (..........+ 10) (x + .......)

p = 3 dan q =10

= (......... + 1) (3x + ........)


13


1)

Tentukanlah faktor dari bentuk aljabar berikut ini! a) x2 + 7x + 12

b) x2 - 6x + 8

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b) x2 - 8x – 9

d) x2 - x – 2

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Tentukanlah faktor dari bentuk aljabar berikut ini! a) 2x2 + 5x + 3

b) 6a2 + 7a + 2

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

14

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= a). Menyelesaikan operasi penjumlahan dan pengurangan bentuk pecahan dengan penyebut suku satu dan suku dua b). Menyelesaikan operasi perkalian dan pembagian bentuk pecahan dengan penyebut suku satu dan suku dua

Tujuan Pembelajaran = Siswa dapat menyelesaikan operasi hitung bentuk pecahan aljabar dengan penyebut suku satu dan suku dua

Penjumlahan dan pengurangan bentuk pecahan aljabar Pada penjumlahan dan pengurangan pecahan, penyebutnya harus sama. Jika penyebut-penyebutnya berbeda harus disamakan dahulu dengan cara mencari Kelipatan Persekuuan ter-Kecil (KPK) dari penyebut-penyebutnya.

a c ad  bc   b d bd

1)

Sederhanakan penjumlahan bilangan pecahan berikut!

1 2  5 3 2)

Penyelesaian:

1 2 3 10     .................................. 5 3 15 15

Sederhanakan penjumlahan dan pengurangan bilangan pecahan berikut!

5 2  3x 5 x 3)

a c ad  bc   b d bd

Penyelesaian:

5 2 25 .......... ............  KPK dari 3x dan 5x adalah 15x     3x 5 x 15 x 15 x 15 x

Sederhanakan penjumlahan dan pengurangan bilangan pecahan berikut!

5 2  3x 5 x

Penyelesaian:

x x  3 ........ 2( x  3)  KPK dari 2 dan 3 adalah 6    2 3 6 6 3x 2 x  ........   6 6 .........  ........  ...........  6 ....x  ......  6


15


Perkalian dan pembagian bentuk pecahan aljabar o Perkalian bentuk pecahan aljabar Misalnya, bentuk pecahan aljabar a/b dan c/d dengan b≠ 0 dan d ≠ 0

a c ..........   b d bd

dengan b≠0 dan d≠0

Coba kamu kerjakan: 1)

3 7  = ............................................................................................... 5 2

2)

3 2  = ............................................................................................... a a

Selesaikan perkalian bentuk pecahan aljabar:

2 3  p q

Penyelesaian:

2 3 2 ...........     ................. p q ............. q

o pembagian bentuk pecahan aljabar Misalnya, bentuk pecahan aljabar a/b dan c/d dengan b ≠ 0, c ≠ 0 dan d ≠ 0

a c a  .......... ............    b d ...........  c ............

dengan b≠0 dan d≠0

Coba kamu kerjakan: 1)

3 2   .............................. 5 3

2)

x y   .............................. 3 6

Selesaikan perkalian bentuk pecahan aljabar

Penyelesaian:

3x 2 3x ........    4 x 4 2

........x 2  ............

16


3x 2  4 x


Selesaikan penjumlahan dan pengurangan bilangan pecahan berikut! a)

b)

1 1  a 2a

e) 5a

x 2  y x

f)

4 5  3b 7ab

g)

3 p  5 2q  2 pq 2 3q

c) 2a  5

6

d)

1)

a7 3

7b 5

a 1 2  a  3a  2 5a

Tentukan hasil perkalian bentuk pecahan aljabar berikut: a) a 6b b) 3 pq

5b

2)



2





7b

2r



4 pr 9q 2

Tentukan hasil pembagian bentuk pecahan aljabar berikut. m m3 a) 2a  3 b)  7b 2b m 1 m 1

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


17




Kompetensi Dasar


Indikator

= a). Menyelesaikan operasi pangkat bentuk pecahan aljabar b). Menyederhanakan bentuk pecahan aljabar

Tujuan Pembelajaran = Siswa dapat menyelesaikan operasi hitung bentuk pecahan aljabar dengan penyebut suku satu dan suku dua

Menyelesaikan operasi pangkat pada bentuk pecahan aljabar

 Sederhanakan bentuk pecahan aljabar 2

 3a   6b       b   3b 

Penyelesaian:

3

2

3

9a 2 8a 3  3a   2a         b 2 ........b 3  b   3b 



........... ...........

Menyelesaikan operasi pangkat pada bentuk pecahan aljabar Pecahan aljabar dapat disederhanakan dengan mengalikan pembilang dan penyebut dari pecahan itu dengan suatu bilangan yang sama yaitu, KPK dari masing- masing penyebutnya.

Sederhanakan pecahan aljabar berikut

2 1  3 4 1 1  4 2

Penyelesaian: 2 1 12 2  1     3 4  3 4 1 1 1 1  12   4 2 4 2



...........  3 3  ...........



......... .........

1) Sederhanakanlah bentuk pecahan aljabar berikut! 2

a)

 2a   3a       3b   b 

3

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

18


“Memahami lebih baik daripada sekadar membaca” 3

b)

 a 2   2a 3      2   3b   b 

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

c)

 2 x10 x 6  3x 2   2   5 4 y  y  3y

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2) Sederhanakanlah pecahan bersusun berikut!

a) 1 1  2 3 1 1 6

c)

2 1 x x 2  2 x

b) 1  1 x 1 1 x

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


19

BAB 2 RELASI DAN FUNGSI Pembelajaran 2.1 Standar Kompetensi


Kompetensi Dasar

= Memahami Relasi dan Fungsi

Indikator

= 1. Menjelaskan pengertian relasi dan fungsi 2. Menyatakan masalah sehari-hari yang berkaitan dengan relasi dan fungsi ke dalam diagram panah, himpunan pasangan berurutan dan diagram Cartesius

Tujuan Pembelajaran = 1. Siswa dapat menjelaskan pengertian relasi dan fungsi 2. Siswa dapat menyatakan masalah sehari-hari yang berkaitan dengan relasi dan fungsi kedalam diagram panah,himpunan pasangan berurutan dan diagram cartesisus.

Relasi Perhatikan permasalahan berikut! Bu Ani mempunyai empat orang anak yaitu Rina, Siska Dedi dan Tomi. Masing–masing anak mempunyai makanan kegemaran yang berbeda. Rina gemar makan bakso, Siska gemar makan sate dan bakso, sedangkan Dedi dan Toni gemar makan mie goreng. Jika anak–anak bu Ani di kelompokkan dalam suatu himpunan A, maka kita dapat menuliskannya sebagai berikut: A = {...........................,..................................,...............................,.....................................} jenis makanan yang digemari anak-anak bu Ani dikelompokkan dalam suatu himpunan B, maka kita dapat menuliskannya sebagai berikut: B = {........................................,.........................................,................................................} himpunan anak-anak buk ani mempunyai hubungan dengan himpunan jenis makanan yaitu “ kegemaran” Dari permasalahan di atas, maka kita dapat menyimpulkan bahwa: relasi dari himpunan A ke himpunan B adalah : hubungan ..................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

Menyatakan Relasi Relasi yang menghubungkan himpunan yang satu dengan himpunan lainnya dapat disajikan dalam beberapa cara, yaitu diagram panah, diagram Cartesius, dan himpunan pasangan berurutan.

20


“Belajar adalah investasi berharga untuk masa depan”

o Diagram panah Apabila permasalahan Bu Ani seperti dinyatakan dengan diagram merepresentasikan sebagai berikut:

panah, maka kita dapat

o Himpunan Pasangan Berurutan Apabila diagram panah pada nomor (1) dinyatakan dengan pasangan berurutan maka dapat ditulis sebagai berikut: Himpunan pasangan berurutan = {(Rina,................), (.................., bakso), (................., (.................,...................), (.................,...................)}

...................),

o Diagram Cartesius Dari himpunan pasangan berurutan pada no (2) apabila dinyatakan dalam diagram Cartesius, maka grafiknya dapat digambar disamping.

1)

Tuliskan sebuah contoh relasi yang terjadi dalam kehidupan sehari-hari dan nyatakan dalam diagram panah, himpunan pasangan berurutan dan diagram Cartesius: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Himpunan P = {6, 10, 14, 22, 26} dan Q = {7, 11, 13, 3, 5}, tentukan: a) Relasi yang mungkin dari himpunan P ke himpunan Q b) Nyatakan relasi tersebut dalam diagram panah, diagram Cartesius, dan himpunan pasangan berurutan! ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

BAB 2 RELASI DAN FUNGSI

21


3)

Perhatikan diagram Cartesius berikut! Ceritakanlah dengan bahasa diagram Cartesius disamping!

kamu

tentang

.............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Relasi dari A = {a, e, i, o, u} ke B = {b, c, d, f, g, h} dinyatakan sebagai R = {(a,b), (a,c), (e,f), (i,d ), (o,g), (o,h), (u,h)}. Nyatakan relasi tersebut ke dalam bentuk diagram panah dan diagram Cartesius ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

5)

Tentukan relasi yang memenuhi dari diagram tersebut, kemudian nyatakan dalam diagram panah dan himpunan pasangan berurutan.

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

22

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= 1. Menjelaskan pengertian pemetaan/fungsi 2. Menyatakan masalah sehari-hari yang berkaitan dengan pemetaan/fungsi ke dalam diagram panah, himpunan pasangan berurutan dan diagram Cartesius 3. Menyebutkan domain, kodomin dan range suatu fungsi

Tujuan Pembelajaran = 1. Siswa dapat menjelaskan pengertian pemetaan/fungsi 2. Siswa dapat menyatakan masalah sehari-hari yang berkaitan dengan pemetaan/fungsi kedalam diagram panah, himpunan pasangan berurutan dan diagram Cartesius. 3. Siswa dapat Menyebutkan domain, kodomin dan range suatu fungsi

Pengertian Fungsi Kamu sudah mengetahui atau memahami relasi, untuk memahami pengertian fungsi atau pemetaan. Perhatikan beberapa contoh relasi berikut.

dari contoh–contoh relasi diatas, Gambar (1) dan Gambar (2) merupakan fungsi atau pemetaan. Gambar (2) dan Gambar (4) bukan merupakan fungsi. Coba kamu jelaskan perbedaan relasi dan fungsi:

Relasi adalah : .................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... Fungsi adalah : ................................................................................................................................... ............................................................................................................................................................... ...............................................................................................................................................................

Menyatakan fungsi Fungsi dapat dinyatakan dalam: diagram panah, himpunan pasangan berurutan, dan diagram Cartesius


23


o Diagram panah

Pada relasi diatas: himpunan A={ ...................,.......................,....................... } disebut daerah asal (domain) himpunan B={ .....................,.......................,...................... } disebut daerah kawan (kodomain) Sedangkan range atau daerah hasil adalah { ...........,............,............... }

o Himpunan Pasangan Berurutan Diagram panah pada nomor (1) dinyatakan dengan pasangan berurutan maka dapat ditulis sebagai: ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

o Diagram Cartesius Himpunan pasangan berurutan pada no (2) apabila dinyatakan dalam diagram Cartesius, maka grafiknya dapat digambar disamping:

1)

Perhatikan gambar (i), (ii), dan (iii), manakah yang merupakan fungsi (pemetaan) dan bukan fungsi, serta berikan alasannya!

Penyelesaian i) …………………………………… ii) …………………………………… iii) …………………………………… 2)

24

Bentuklah kelompok yang terdiri atas 2 orang, Cari dan amati kejadian-kejadian di lingkungan sekitarmu. Tulislah hal-hal yang termasuk fungsi sebanyak 2 buah. Lalu sajikan hasil temuanmu dalam diagram panah, diagram Cartesius, dan himpunan pasangan berurutan.


“Belajar adalah investasi berharga untuk masa depan” ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Perhatikan diagram Tentukanlah: (i) domain; (ii) kodomain; (iii) range;

panah

pada

Gambar

disamping.

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


25




Kompetensi Dasar


Indikator

= Menentukan banyaknya pemetaan dua himpunan A dan B.

Tujuan Pembelajaran = Siswa dapat menetukan banyaknya pemetaan dua himpunan A dan B Untuk menentukan banyaknya pemetaan yang mungkin dari dua himpunan, kerjakan kegiatan berikut: 1)

Jika himpunan A = {a,b} maka n(A) = 2 Himpunan B = {p} maka n(B)= 1 carilah banyaknya pemetaan yang mungkin terjadi dari A ke B. n(A)=2 dan n(B)=1 Banyak pemetaan dari himpunan A ke himpunan B yang mungkin terjadi = 1 cara.

2)

Misalkan A = {a,b} maka n(A) = …… B = {p,q} maka n(B) = …… Buat diagram panah pemetaan yang mungkin dari A ke B

banyak pemetaan dari himpunan A ke himpunan B yang mungkin terjadi = ................. 3)

Misalkan A = {a,b} maka n(A) = …… B = {p,q, r} maka n(B) = …… Buat diagram panah pemetaan yang mungkin dari A ke B

banyak pemetaan dari himpunan A ke himpunan B yang mungkin terjadi = ................. 4)

Misalkan A = {a,b,c} maka n(A) = …… B = {p} maka n(B) = ……

26



Buat diagram panah pemetaan yang mungkin dari A ke B

banyak pemetaan dari himpunan A ke himpunan B yang mungkin terjadi = ................. dari hasil diatas, isilah tabel berikut dan analisalah untuk mendapatkan nilai yang lain ! n(A)

n(B)

Banyak pemetaan dari A ke B

Kuadrat

2

1

1

12

2

2

4

………………..

2

3

9

32

2

4

……………….

……………..

2

5

……………….

……………….

3

1

1

13

3

2

8

23

4

1

……………….

……………..

4

2

……………….

……………….

k

l

……………….

…………………

Berdasarkan tabel di atas dapat di simpulkan bahwa: jika n(A) = a dan n(B) = b maka banyak semua pemetaan yang mungkin dari A ke B adalah .............................................

1)

Diketahui himpunan A={a,b,c,d,e} dan himpunan B={p, q}. Berapakah banyak pemetaan yang terjadi dari himpunan A ke himpunan B tersebut? ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

dari soal nomor (1) tentukan pula banyaknya pemetaan darihimpunan B ke himpunan A. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

P={huruf vokal} dan Q={bilangan asli kurang dari 4}.tentukan banyaknya pemetaan yang mungkin dari himpunan P ke himpunan Q! ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


27




Kompetensi Dasar

= Memahami Relasi dan Fungsi.

Indikator

= 1. Mempresentasikan suatu fungsi dalam notasi. 2. Menghitung nilai fungsi.

Tujuan Pembelajaran = 1. Siswa dapat mempresentasikan suatu fungsi dalam notasi. 2. Siswa dapat menghitung nilai fungsi.

Notasi Fungsi Fungsi dinotasikan dengan huruf kecil, seperti f, g, atau h. Pada fungsi f memetakan x anggota himpunan A ke y anggota himpunan B, dinotasikan dengan f:x→f (x)

Perhatikan gambar berikut. Pada Gambar disamping menunjukkan: fungsi himpunan A ke himpunan B menurut aturan f :x  2x + 1.

x merupakan anggota domain f. Fungsi f :x  ........................ berarti fungsi f memetakan x ke ...................... Oleh karena itu, bayangan x oleh fungsi f adalah ......................... Jadi, dapat dikatakan bahwa f (x) = ................................ adalah rumus untuk fungsi f. Jika fungsi f :x  2x+1 dengan x anggota domain f, rumus fungsi f adalah:

f (x) = .................................................

1)

Fungsi f : x → 4x+1 rumus fungsinya adalah .......................................................................................................................................................................... ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Fungsi g : x → rumus fungsinya adalah .......................................................................................................................................................................... ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Fungsi h : x → 2x2-1 rumus fungsinya adalah .......................................................................................................................................................................... ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

28



Menghitung Nilai Fungsi Perhatikan contoh berikut ini.

Y= f(x) = x + 2 x adalah variabel bebas dan y adalah variabel terikat. Pada materi yang akan kamu terima sekarang adalah menghitung nilai fungsi. Menghitung nilai fungsi berarti mensubstitusikan nilai variabel bebas ke dalam rumus fungsi sehingga diperoleh nilai variabel bergantungnya.

Perhatikan soal berikut dan cobalah diskusikan cara menyelesaikannya dengan temanmu. 1)

Pemetaan f:G→R ditentukan oleh f(x)=x +2 dengan G={-1,0,1,2,3,4} dan R adalah himpunan bilangan real. maka daerah hasil dari f adalah ..................................

Penyelesaian: f(x)

= x+2

Substitusikan setiap anggota domain G ke rumus fungsi f(x) didapat:

f(-1)

= (................) + 2

f (2)

= ......................

f (3)

= ......................

= ...................... f(0)

= (................) + ................... = ......................

f (1) =

......................

f (4) = ......................

Daerah hasil dari f adalah { ................................................ } Bayangan -1 oleh K adalah ........................................... 2)

Jika f(x) = -3 maka nilai x adalah .........................................

Penyelesaian: f(x)

= 3

x+2

= -3

x 3)

= ....................................

Fungsi f pada R ditentukan oleh formula f(x) = ax + b dan diketahui f(4) = 6 dan f(2) = -2. Tentukan bentuk fungsi f.

Penyelesaian: f(x)

= ax + b

f(4)

= 6 → a (..........) + b

= 6

f(2)

= -2 → a (..........) + b

= -2 (-)

.............. + .............

= 6

.....................

= ..........................

.....................

= ..........................

Substitusikan a =.................... Ke persamaan 2a + b = -2 maka akan diperoleh:

2 (..............) + b b

= -2 = ........................

Karena a = ........................ dan b = ..........................


29


Jadi fungsi f adalah f(x) = ax + b

= ................... + ...................

Suatu pemetaan K ditentukan oleh K : x → 3x – 1 dengan x anggota bilangan real. Tentukan: a) Bayangan 2 oleh K ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b)

Nilai k untuk x = -4 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

c)

Nilai r sehingga k(r) = 7 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

30

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= Menentukan bentuk fungsi jika nilai dan data fungsi diketahui

Tujuan Pembelajaran = Siswa dapat menentukan bentuk fungsi jika nilai dan data fungsi diketahui. Suatu fungsi dapat ditentukan rumusnya jika nilai data diketahui. Bagaimanakah caranya? Untuk menjawabnya, pelajarilah contoh soal berikut.

Fungsi h pada himpunan bilangan Real ditentukan oleh rumus h(x) = a x + b, dengan a dan b bilangan bulat. Jika h (–2) = –4 dan h (1) = 5, tentukan: a) nilai a dan b, b) rumus fungsi tersebut.

Penyelesaian: h(x) = ax + b a)

Oleh karena

h(–2)

= –4

h(–2)

= a (..............) + b = –4

,maka

...................a + b = –4 h(1)

= 5

h(1)

= a (...............) + b = 5

---------------------------------- persamaan (1)

maka

..................... + b = 5 b = ................

---------------------------------- persamaan (2)

Substitusikan persamaan (2) ke persamaan (1), diperoleh: ..............a + b

= –4

...............a + (.............)

= –4

..............a + 5 – a

= –4

–3a + 5

= –4

–3a

= –9

a

= 3

Substitusikan nilai a =3 ke persamaan (2), diperoleh : b

= 5–a

b

= 5 – .....................

b

= .......................

Jadi, nilai a sama dengan 3 dan nilai b sama dengan .................................. b)

Oleh karena nilai a = 3 dan nilai b = ..................... rumus fungsinya adalah h(x) = ............... x + .......................


31


1)

Fungsi f ditentukan oleh f(x) = ax + b. Jika f(2) = 12 dan f(–3) = –23, tentukanlah: a) nilai a dan b, ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

b)

rumus fungsi tersebut.

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Diketahui R(x) = ax + b, jika R(-2) = -4 dan R(-6) = 12 tulislah bentuk fungsi R. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

32

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar

= Membuat sketsa grafik fungsi aljabar sederhana pada sistem koordinat Cartesius

Indikator

= 1. Menentukan koordinat suatu titik pada koodinat Cartesius 2. Membuat gambar grafik pada koordinat coordinat Cartesius dari persamaan yang ditentukan

Tujuan Pembelajaran = 1. Siswa dapat menggambar titik coordinat Cartesius dan menentukan titik koordinat pada koordinat Cartesius 2. Membuat gambar grafik pada koordinat coordinat Cartesius dari persamaan yang ditentukan

Menggambar grafik fungsi Sebelum kamu membuat grafik fungsi pada kooradinat Cartesius, terlebih dahulu kamu memahami: 1) unsur unsur yang ada pada koordinat Cartesius 2) Menggambarkan titik koordinat Cartesius

1) Jawablah pertanyaan dibawah ini. a) Dari gambar disamping, garis horizontal (mendatar) disebut dengan sumbu .................... dan garis tegak (....................) disebut sumbu Y b) Sumbu mendatar (......... disebut absis Sumbu tegak (y) disebut ....................................... , sedangkan Pasangan absis dan ordinat (....................) disebut koordinat c) Perhatikan koordinat titik P merupakan pasangan 3 dan 4 ditulis (……… , ………), 3 disebut …………………… dan 4 disebut …………………… d) Koordinat titik A (………… , …………), dan koordinat titik B (………… , …………)

2) Diketahui koordinat titik P (3,4), Q (-3,4) dan titik R (2,-3), gambarkanlah titik tersebut pada koordinat Cartesius.


33


3) Diketahui suatu fungsi f(x) = 2x + 1 dimana x bilangan Real, Gambar grafik fungsi tersebut.

Penyelesaian: Untuk mengambar grafik grafik fungsi, tentukan daerah asal misal {1,2,3,4, dan 5}, Langkah 1: Tentukan titik koordinat. (dapat disajikan bentuk tebel)

dalam

X

1

2

3

4

5

2x+1

……

…..

7

…..

…..

(x,y)

……

…..

(3,7)

……

…..

Langkah 2: Gambarkan titik koordinat pada gambar disamping Langkah 3: Hubungkan titik pada koordinat Cartesius pada langkah 2, untuk memperoleh grafiknya 4) Apabila suatu fungsi f yang dirumuskan sebagai f(x) = 2x–3 dengan daerah asal A={-2, -1, 0, 1, 2}. a) Tentukanlah dareah hasil atau range dari fungsi f(x) = 2x – 3 b) Tentukanlah letak titik-titik tersebut pada koordinat Cartesius. c) Gambarlah suatu garis yang melalui titik-titik tersebut.

Penyelesaian: a) Daerah hasil atau range dari f(x) = 2x – 3 adalah f(-2) =

2(.................) – 3 = ..............

f(-1) = .......................... f(0)

= ..........................

f(1)

= ..........................

f(2)

= ..........................

Daerah hasil atau range = (............., ................., .............................................) nilai fungsi yang diperoleh dari f(x) = 2x – 3 dapat disajikan pada tabel berikut ini: x -2 -1 0 2 1 2x – 3

...................

(x, y)

(-2, .............)

...............

-3

.................

....................

(.........., ..........)

(0, -3)

(.........., ..........)

(.........., ..........)

b) Letak titik-titik pada poin (a) dapat digambarkan pada koordinat Cartesius berikut ini:

34



c) Untuk menggambar garis dari fungsi f(x) = 2x – 3 yaitu dengan menghubungkan titik-titik yang diperoleh pada poin (b) e)

Apabila suatu fungsi g : x – 3x + 2 dengan daerah asal A = {x1 ≤ x ≤ 5, x bilangan real}. a) Tentukanlah dareah hasil atau range dari fungsi g tersebut. b) Tentukanlah letak titik-titik tersebut pada koordinat Cartesius. c) Gambarlah suatu garis yang melalui titik-titik tersebut.

Penyelesaian: a) Daerah hasil atau range dari g adalah g(1) = ....................................... g(2) = ....................................... g(3) = ....................................... g(4) = ....................................... g(5) = ....................................... nilai fungsi yang diperoleh dari g(x) = – 3x + 2 dapat disajikan pada tabel berikut ini: x 1 2 3 4

5

...................

...................

...............

.................

.................

....................

(x, y)

(.........., ..........)

(.........., ..........)

(.........., ..........)

(.........., ..........)

(.........., ..........)

b) Letak titik-titik pada poin (a) dapat digambarkan pada koordinat Cartesius berikut ini:

c) Untuk menggambar garis dari fungsi f(x) = 2x – 3 yaitu dengan menghubungkan titik-titik yang diperoleh pada poin (b)


35


1)

Apabila suatu fungsi f yang dirumuskan sebagai f(x)=3x–2 dengan daerah asal A={-2, -1, 0, 1, 2}. a) Tentukanlah dareah hasil atau range dari fungsi f(x) = 3x – 2 b) Tentukanlah letak titik-titik tersebut pada koordinat Cartesius. c) Gambarlah suatu garis yang melalui titik-titik tersebut. ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ...............................................................................................................

2)

Diketahui suatu fungsi g dengan rumus g(x) = -5x + 1 dengan daerah asal A = {xl-5 ≤ x ≤ 5, x bilangan real}. a) Tentukan daerah hasil fungsi g. b) Gambarlah grafik fungsi g pada koordinat Cartesius. c) Berupa apakah grafik fungsi g? ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ...............................................................................................................

3)

Diketahui suatu fungsi f dengan daerah asal A = {-2, 2, 5, 7} dengan rumus fungsi f(x)=2x+3 a) Tentukan f(-2) , f(2), f(5) dan f(7). Kesimpulan apa yang dapat kamu peroleh? b) Buatlah tabel fungsi di atas. c) Tentukan daerah hasilnya. d) Gambarlah grafik fungsi dalam koordinat Cartesius. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

36



4)

Diketahui suatu fungsi g dengan daerah asal P = { x l x ≥ 3, x bilangan real} dengan rumus fungsi g(x) = 3x + 4. a) Buatlah tabel fungsi di atas dengan mengambil beberapa nilai x. b) Tentukan daerah hasilnya. c) Gambarlah grafik fungsi dalam koordinat Cartesius. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. .............................................................................................................. ..............................................................................................................

5)

Perhatikan grafik fungsi f pada koordinat Cartesius berikut. a)

Tentukan daerah hasil fungsi f.

b) Tentukan nilai fungsi f untuk x = 0, x = 1, x = 2, x = 3 dan x = 4. c)

Pola apakah yang kamu peroleh?

d) Tentukan rumus fungsi f berdasarkan (b)?

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


37

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

6)

Diketahui suatu fungsi f dengan rumus f(x)=3x–1 dengan daerah asal k {-3, -1, 1, 3, 5, 7} a) Buatlah tabel nilai fungsi f. b) Tentukan daerah hasil fungsi f. c) Gambarlah grafik fungsi tersebut. ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ...............................................................................................................

Nilai Kognitif

38

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua

BAB 3 PERSAMAAN GARIS LURUS Pembelajaran 3.1 Standar Kompetensi

= Memahami bentuk aljabar, relasi, fungsi, dan persamaan garis lurus.

Kompetensi Dasar

= Menentukan gradien, persamaan dan grafik garis lurus.

Indikator

= 1. Mengenal persamaan garis lurus dalam membangun bentuk dan variabel. 2. Menggambar grafik pada bidang Cartesius

Tujuan Pembelajaran = 1. Siswa mengerti persamaan garis lurus dalam membangun bentuk dan variabel. 2. Siswa dapat menggambar grafik pada bidang Cartesius

Sistem koordinat Cartesius Pada materi sebelumnya, kamu telah mempelajari sistem Cartesius, untuk mengingat kembali perhatikan gambar di bawah ini.

Gambar diatas merupakan denah perkemahan pramuka dan daerah yang harus mereka jelajahi untuk kegiatan “mencari jejak”. Dapatkah kamu melengkapi cerita berikut ini? Para kelompok pramuka tersebut terbagi menjadi empat kelompok. Masing-masing kelompok menempati satu tenda, yaitu tenda 1 pada koordinat (2,0), tenda 2 di (........,........), tenda 3 di (........,........), dan tenda 4 di (........,........). Koordinasi setiap kegiatan dilakukan di posko utama, yaitu di (........,........).

BAB 3 PERSAMAAN GARIS LURUS

39


Sebelum kegiatan “mencari jejak” dilakukan, mereka diingatkan untuk mengikuti setiap petunjuk yang diberikan di setiap pos, yaitu pos 1 di (........,........), pos 2 di (........,........), dan pos 3 di (........,........). Mereka juga dilarang masuk ke hutan, yaitu di (........,........) karena sangat berbahaya. Selain itu, mereka juga harus berhati-hati saat melewati tanah lapang yang cukup luas di (........,........), pemakaman di (........,........), dan sungai di (........,........). Para anggota pramuka itu juga harus berusaha mencari adan memecahkan teka-teki yang disembunyikan di (........,........). Dari kegiatan diatas, kamu tentunya sudah semakin lancar membaca koordinat Cartesius.

Menggambar garis lurus pada bidang Cartesius Jika diketahui sebuah pemetaan

f(x)=2x

dengan daerah asal 0x3; xXR. Tentu kamu telah dapat menggambar grafik fungsinya bukan! x

0

1

2

3

2x (x,y) Dalam permasalahan tersebut, persamaan f(x)=2x dapat diubah menjadi y=2x. Untuk menggambar sebuah garis, kamu dapat mengikuti langkah berikut ini: 1) Tentukan minimal dua titik yang memenuhi persamaan tersebut. Pilihlah titik yang memudahkan dalam perhitungan. 2) Buatlah tabel untuk mempermudah perhitungan. 3) Gambarkan titik tersebut pada bidang koordinat Cartesius. 4) Hubungkan titik-titik tersebut. Agar kamu lebih memahaminya, lakukanlah kegiatan berikut ini.

Menggambar garis dengan persamaan y=mx 1)

Gambarkan grafik y = 3x

Penyelesaian:

Langkah 3

Langkah 1 x = .............  y = 3 ............ y = ............ x = .............  y = 3 ............ y = ............ langkah 2 x y (x,y)

40


“Sukses itu dapat terjadi karena persiapan, kerja keras dan mau belajar dari kegagalan”

2)

Gambarkan grafik y = -4x

Penyelesaian: x = .............  y = -4 .......... y = ............ x = .............  y = -4 .......... y = ............ x y (x,y)

3)

Gambarkan grafik y =

x

Penyelesaian: x = .............  y = ............ y = ............ x = .............  y = ............ y = ............ x y (x,y)

 Apakah grafik garis y = 3x, y = -4x dan y = x melewati titik pangkal (0,0) ? Jawab : ..............................................................................................................................................................................................................  Jika koefisien x dari persamaan garis di atas dilambangkan dengan m, maka persamaan garis yang melewati titik pangkal O (0,0) dan titi (x,y) adalah:

y = ……x

Menggambar garis dengan persamaan y=mx+c 1)

Gambarkan grafik y = 3x+1



41


2)

Gambarkan grafik y = 4x-2


 Apakah grafik garis y = 3x+1 dan y = 4x-2 melewati titik pangkal (0,0) ? Jawab : ..............................................................................................................................................................................................................  Garis y = 3x+1 memotong sumbu y di titik ( .............. , .............. )  Garis y = 4x-1 memotong sumbu y di titik ( .............. , .............. )  Dari kegiatan diatas, apa yang dapat kamu simpulkan? Jika koefisien x = m dan berpotongan dengan sumbu y = c, maka persamaan garis tersebut adalah:

y = ............ x + ............  memotong sumbu y di titik ( 0 , ............ )

Gambarlah grafik garis lurus yang memenuhi persamaan berikut: 1)

y= x

2)

y = -3x

3)

y = -x + 5

4)

y= x-3

42



Menentukan Persamaan Garis yang digambar pada Bidang Koordinat Cartesius Pada pembelajaran yang lalu, kamu telah mempelajari cara menggambar grafik garis jika persamaannya diketahui. Sekarang, kamu akan mempelajari hal yang sebaliknya, yaitu menentukan persamaan garis jika gambar garisnya diketahui. Untuk itu, analisa gambar tersebut!

Apakah garis melalui titik (0,0)

YA

TIDAK

persamaan

persamaan

y = mx

y = mx + c

Pilih salah satu titik selain (0,0) untuk menentukan nilai m

Pilih 2 titik sembarang untuk menentukan nilai m dan c

Untuk lebih memahaminya, lakukanlah kagiatan berikut ini. 1) Tetukan persamaan garis dari gambar dibawah ini. Langkah-langkah:  Apakah garis melalui titik (0,0)? Jawab: .......................  Maka persamaan garisnya adalah: y = .......................  Ambil satu titik pada garis. misalkan ( 4, .......... )

maka:

x = ....................... y = ........................

 substitusikan nilai x dan nilai y y =m x .................

= m ( ................. )

m = .................  jadi persamaan garisnya adalah y = ..............................


43


2) Tetukan persamaan garis dari gambar dibawah ini. Langkah-langkah:  Apakah garis melalui titik (0,0)? Jawab: .......................  Maka persamaan garisnya adalah: y = .......................  Titik potong dengan sumbu y dititik ( 0, .......... ) maka

c = ........................

 Titik potong pada sumbu x adalah: dititik ( .......... , 0 )

maka:

x = ....................... y = ........................

 substitusikan nilai x, nilai y dan nilai c y =m x + c .................

= m ( ................. ) + .................

m = .................  jadi persamaan garisnya adalah y = ............... + ...............

1) Tentukan persamaan garis a, b, dan c pada gambar dibawah ini.

44



2) Tentukan persamaan garis a, b, dan c pada gambar dibawah ini.

3) Gambar garis yang melalui titik pangkal (0,0) dan titik (4,-3). Tentukanlah persamaan garisnya.

4) Gambar garis yang melalui titik P (0,2) dan Q (2,0). Kemudian, tentukanlah persamaan garisnya.

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


45




Kompetensi Dasar


Indikator

= 1. Menjelaskan pengertian gradien. 2. Menggambar suatu garis yang melalui titik pusat dan titik lain yang diketahui pada koordinat Cartesius. 3. Menentukan gradien dari suatu garis yang melalui titik pusat dan titik lain yang diketahui pada koordinat Cartesius.

Tujuan Pembelajaran = 1. Menjelaskan pengertian gradien. 2. Menggambar suatu garis yang melalui titik pusat dan titik lain yang diketahui pada koordinat Cartesius. 3. Menentukan gradien dari suatu garis yang melalui titik pusat dan titik lain yang diketahui pada koordinat Cartesius.

Pengertian Gradien Pernahkah kamu melalui jalan yang naik dan turun seperti halnya di daerah pegunungan? Lereng gunung memiliki kemiringan tanah yang tidak sama, ada yang curam ada juga yang landai. Oleh karena itu, pembangunan suatu jalan yang menanjak dan berkelok-kelok seperti di pegunungan diperlukan perhitungan tertentu agar kendaraan mudah melewatinya. Salah satu perhitungan matematika yang harus diperhatikan dalam pembangunan jalan seperti itu adalah kemiringannya. Tingkat kemiringan inilah yang disebut gradien. Untuk memahami persoalan tersebut, maka perhatikanlah gambar disamping (kanan atas)! Apabila tanjakan mobil pada gambar yang tampak seperti permasalahan di atas kita sajikan pada grafik pada Gambar disamping (kanan bawah)! Kemiringan jalan merupakan perbandingan garis tegak (vertikal) dengan garis mendatar (horizontal). a) Tentukan panjang garis tegak dengan cara menghitung banyaknya petak satuan. Jadi, banyak petak satuan pada garis tegak adalah ………………………………………………………….. b) Tentukan panjang garis mendatar dengan cara menghitung banyaknya petak satuan. Jadi, banyak petak satuan pada garis mendatar adalah ………………………………………………

46



Sehingga diperoleh kemiringan jalan adalah

panjang garis tegak ................................  panjang garis mendatar ................................. ................................ .................................

Kemiringan jalan disebut gradien, maka gradiennya adalah

Pada grafik Cartesius berikut terdapat garis OA, garis OB dan garis OC. Tentukanlah gradien masingmasing garis tersebut!

a) Gradien garis OA =

b) Gradien garis OB =

c) Gradien garis OC =

panjang gari s tegak OA panjang gari s mendatar OA



komponen y komponen .......

.............................................. panjang gari s mendatar OB





-3 .....

komponen ....... komponen x



............ .............

.............................................. .............................................. ............   .............................................. .............................................. .............

Berdasarkan perhitungan gradient garis diatas dapat disimpulkan 1) Gradient positif menyatakan kemiringan garis ke kanan 2) Gradient negative menyatakan kemiringan garis ke.................................. Gradien suatu garis yang melalui titik O(0,0) dan titik (x,y)

m

komponen y komponen .......

atau

m

y ...............


47


Gradien Garis y = mx Perhatikan gambar di bawah ini! Garis-garis pada gambar di samping melalui titik pangkal koordinat. Hubungan antara persamaan garis dengan gradiennya ditunjukkan pada table berikut. Persamaan Garis

Gradien

y= x

=

y = 2x

=…

Y = ….

=…

Dari tabel di atas terlihat ba hwa koefisien x dari suatu persamaan garis ternyata merupakan .....................................................................

garis itu.

Misalkan: Persamaan garis y =

x mempunyai gradien

dan persamaan garis y = 2x mempunyai gradien …….,

Sehingga dengan demikian dapat diambil suatu kesimpulan: Persamaan garis y = mx mempunyai gradien ...................

gradien Garis y = mx + c Perhatikan gambar disamping! Dari gambar di atas terlihat sketsa grafik persamaan garis:

y = 4x + 3 dan y = 2x – 4 Perhatikan table berikut Persamaan Gradien Garis

Titik potong

y = 3x + 3

m = _______ = .................

(.............., 3 )

y = 2x – 4

m = _______ = .................

(.........., ..........)

dengan demikian dapat diambil suatu kesimpulan : Persamaan garis

y = mx+c mempunyai gradien ...................

dan memotong sumbu y titik (0, ....... )

48



Gradien Garis ax + by + c = 0 Dalam menentukan gradien garis yang berbentuk ax + by + c = 0, maka kita harus mengubahnya ke dalam bentuk y = mx + c ax + by + c = 0

by = - ax – c

y=

x –

Berdasarkan penjabaran di atas dapat kita lihat bahwa gradien dari bentuk persamaan garis ax + by + c = 0 adalah m = -

dan memotong sumbu y di (0 , -

𝐜

)

 Tentukan gradien garis 3x+ 6y + 10 = 0

Penyelesaian : 3x + 6y + 10 = 0 berarti a = ............., b = 6 dan c = ............. Jadi, gradien m = -

= -

= -

Ingat kembali menentukan letak suatu titik pada koordinat Cartesius! 1) Tentukanlah letak titik-titik P (0,0), Q(4,2), R(0,3), S(-3,1), T(-5,-2) a) Gambarkan suatu garis yang melalui titik PQ, PR, PS, dan PT. b) Tentukalah gradien dari garis PQ, PR, PS, dan PT.

Penyelesaian : ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................ ................................................................................................


49


2) Tentukan gradien garis yang melalui titik pusat dan titik. a)

P(2,6)

b) Q(4,-8)

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3. Tentukan gradien dari masing-masing persamaan garis berikut. a) 2x – 6y + 7 = 0 b) -3x – 6y – 4 = 0 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4. Tentukan gradien garis berikut. a) Y= 8x b) Y-7x = 0 c) Y=-5x-2 d) X+2y= 0 e) 3x-y= 0 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

50

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= 1. Menentukan gradien suatu garis yang melalui dua titik yang diketahui . 2. Menentukan gradien suatu garis yang sejajar sumbu x dan y 3. Menentukan gradien garis apabila diketahui kedudukan dua buah garis: a. Saling sejajar b. Saling tegak lurus.

Tujuan Pembelajaran = 1. Siswa dapat menentukan gradien suatu garis yang melalui dua titik yang diketahui 2. Siswa dapat menentukan gradien suatu garis yang sejajar sumbu x dan y 3. Siswa dapat menentukan gradien suatu garis apabila diketahui kedudukan dua buah garis: a. Saling sejajar b. Saling tegak luruspada koordinat Cartesius.

Menentukan gradient yang melalui dua titik kerjakanlah kegiatan berikut: 1)

Tentukanlah letak titik A(-3,1) dan titik B(2,4) pada koordinat Cartesius a) Gambarkan garis yang melalui titik A dan titik B tersebut b) Tentukanlah gradien dari garis tersebut.

Penyelesaian:

y y komponen y 4  .......... .. .......... ...... B A m     AB .......... .  .......... .......... ( 3) .......... ...... komponen x y  y komponen y 1  .......... .. .......... ...... A B m     BA komponen x .......... .  .......... .......... .......... .......... ......

Apakah yang dapat kamu simpulkan tentang gradien garis AB dan gradien garis BA? ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Jika titik A(x1,x2) dan B(x1,y2) maka gradien yang melalui dua titik adalah Gradient m=

𝐲𝟐 −𝒚𝟏 𝐱 𝟐 −𝒙𝟏


51


Gradien Garis yang Sejajar Sumbu x

Diskusikan dengan temanmu bila gradien suatu garis adalah 0, bagaimanakah kedudukan garis tersebut?

Gradien Garis yang Sejajar Sumbu y

Diskusikan dengan temanmu bila gradien suatu garis tidak didefenisikan, bagaimanakah kedudukan garis tersebut?

Gradien Garis-garis yang Saling Sejajar. Perhatikan gambar disamping. Garis AB dan garis CD merupakan garis-garis yang saling sejajar. Gradien pada garis-garis sejajar juga dapat kamu tentukan dengan menggunakan rumus yang sama pada masingmasing garis tersebut!

gradien garis AB mAB

= ______________

Gradian garis CD = ______________

= ______________

mCD = ______________

= ...........................

52


= ...........................


Dari contoh gambar di atas, ternyata kedua garis mempunyai gradien yang ..........................................., sehingga dapat diambil kesimpulan bahwa : Garis-garis yang sejajar memiliki gradien yang ..................................................... Atau, Jika garis-garis memiliki gradien yang ..................................................... Maka pastilah garis-garis tersebut saling.....................................................

Gradien Garis-garis yang Saling Tegak Lurus Pada gambar di samping, garis p dan garis q saling tegak lurus. Sehingga untuk menentukan gradien pada garis tegak lurus dapat kamu tentukan dengan menggunakan rumus yang sama dengan cara mengambil dua titik pada masing-masing garis tersebut. gradien garis mP =

mTR

Gradian garis mq =

mTS

= ______________

= ______________

= ________

= ________

mp x mq = ____________ x ______________ = ...................................... Dari kedua contoh pada gambar di atas, ternyata hasil kali gradien-gradiennya adalah ………, sehingga dapat diambil kesimpulan bahwa: Hasil kali gradien-gradien garis-garis yang saling tegak lurus adalah .........................................

1 ) Suatu garis p bergradien - 4. Tentukan gradien garis q bila garis q: a) Sejajar dengan garis p b) Tegak lurus dengan garis p …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… 2) Suatu garis melalui titik P(-6,8) dan Q(4,-7). a) Hitunglah . b) Jika garis k tegak lurus dengan PQ, tentukan gradien garis k. …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………………………………………………………………………… ……………………………………………………………………………………………………………………………………………………………………………………


53


3) Tentukan persamaan garis yang melalui titik a) A(–2, 3) dan sejajar garis y = –x – 5

b) E(2, 4) dan sejajar garis x = 3y + 3

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4) Tentukan persamaan garis yang melalui titik (2, 5) dan tegak lurus dengan garis berikut: a) 2x + y + 5 = 0

b)

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Nilai Kognitif

54

Nilai Afektif

Nilai Psikomotorik


y

1 x6 2

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Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= Menentukan persamaan garis lurus jika: a. Melalui dua titik. b. Melalui satu titik dengan gradien tertentu

Tujuan Pembelajaran = Siswa dapat menentukan persamaan garis lurus jika: a. Melalui dua titik. b. Melalui satu titik dengan gradien tertentu

Menentukan Persamaan Garis jika diketahui gradien m dan suatu titik pada garis Misalkan persamaan garis y =mx + c dan P1(x1 , y1) pada garis tersebut. Untuk x = x1 dan y =y1 diperoleh: y1 = m x1 + c

atau

c = y1 – …… c = y1 –……

disubstitusikan pada y = mx + c menjadi: y = mx + (y1 - ….) y = m x – m x1 + ……. diperoleh rumus : y – y1 = m (x – x1 )

Tulislah persamaan garis yang memiliki gradien –2 dan memotong titik (4, 10)!

Penyelesaian: Untuk menjawab soal tersebbut, dapat di kerjakan dengan dua cara berbeda, yaitu: Cara 1 gradien garis adalah- 2 ; m=-2 ; x1= 4 ; dan y1= 10, maka digunakan rumus : y – y1

= m(x – x1 )

y – ……

= -2(x – ………)

y – ……

= ……..+ ………

y

= ……………

y

= ……………

Cara 2 gradien garis adalah –2 , sehingga nilai m = -2 Titik (4, 10) diperoleh x=4 dan y= 10. substitusikan pada persamaan y=mx + c untuk mengetahui nilai c y …… 10 c

= mx + c = -2(………) + c = ………… + c = ……………

Jadi persamaan garis yang dimaksud adalah y = ................................................

Persamaan Garis yang Melalui Dua Titik Sebarang (x1, y1) dan (x2, y2) Dengan memperhatikan bahwa gradien yang melalui titik A(x 1,y1) dan B(x2,y2) adalah: y y 1 m  2 AB x x 2 1


55


Maka persamaan garis yang melalui titik A dan B adalah:



y y 1 xx y-y  2 1 1 x x 2 1



Atau dapat disimpulkan bahwa:

Tentukan persamaan garis yang melalui titik (3, –5) dan (–2, –3).

Penyelesaian: Dengan menggunakan rumus, substitusikan titik (3,-5) dan (-2,-3) ke persamaan: yy xx 1  1 y y x x 2 1 2 1

y  ........... . x  ...........  ......... .......... ........... ..  ........... .. .......... .......... ... ............ ....



.......... .......... ...... .......... .......

..............................

= ..............................

..............................

= ..............................

..............................

= ..............................

..............................

= ..............................

..............................

= ..............................

y

= ..............................

1) Tentukan persamaan garis berikut ini. a) Garis bergradien 4 dan melalui titik (0,-7)

b) Garis bergradien –3 dan melalui titik (0,-5)

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2)

Tentukan persamaan garis yang melalui titik: a) D(3, 0) dan bergradien ½

3)

b) C(7, 1) dan bergradien 1/5

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Tentukan persamaan garis yang melalui titik-titik berikut: a) A(3, –2) dan B(–1, 3)

b) Q(–5, 0) dan R(3, 4)

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c) K(7, 3) dan L(–2, –1) ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ...............................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


57




Kompetensi Dasar


Indikator

= Menggambar grafik garis lurus jika: c. Melalui dua titik. d. Melalui satu titik dengan gradien tertentu e. Persamaan garisnya diketahui

Tujuan Pembelajaran = Siswa dapat menggambar grafik garis lurus jika: c. Melalui dua titik. d. Melalui satu titik dengan gradien tertentu e. Persamaan garisnya diketahui Pada materi sebelumnya kita telah belajar tentang menentukan gradien dari suatu garis lurus dalam berbagai bentuk. Pada materi kita akan mempelajari cara menggambar grafik garis lurus jika:

Grafik Garis Lurus melalui Dua Titik.

 Gambarlah grafik garis lurus yang melalui titik (1,6) dan (3,2)

Penyelesaian : Menggambar grafik garis lurus melalui dua titik dapat dilakukan secara langsung dengan menentukan letak titik-titiknya terlebih dahulu. Setelah itu, kita tarik suatu garis lurus yang menghubungkan kedua titik tersebut. Perhatikan gambar disamping!

Melalui satu titik dengan gradien tertentu

 Gambarlah grafik garis lurus yang mempunyai gradien

58


melalui titik (0,4)


Penyelesaian : Cara menggambar grafik garis lurus melalui satu titik dengan gradien tertentu, terlebih dahulu kita harus menentukan nilai x dan y dari gradien tersebut. m=

=

,

berarti diperoleh: nilai x = .................... dan nilai y = .................... Kemudian letakkan nilai titik x dan y tersebut pada koordinat Cartesius yang dihitung melalui titik yang telah ditentukan, setelah itu hubungkan titik tersebut dengan titik yang telah ditentukan sebelumnya. Perhatikan gambar disamping!

Persamaan garisnya diketahui

 Gambarlah grafik garis lurus yang mempunyai persamaan garis y = 2x  6

Penyelesaian: Persamaan garis y = 2x  6 memotong sumbu x di (0,……)

memiliki Gradien m: m= ................................. sehingga: y = .......................... dan x = .......................... Kemudian, letakkan nilai titik x dan y tersebut pada koordinat Cartesius yang dihitung melalui titik yang telah ditentukan. Setelah itu, hubungkan titik tersebut dengan titik yang telah ditentukan sebelumnya. Kemudian, buatlah grafik garis lurus pada gambar disamping!


59


1)

Gambarlah grafik garis lurus yang melalui titik (-3,-2) dan (5,7) !

2)

Gambarlah grafik garis lurus yang mempunyai gradien -

3)

Gambarlah grafik garis lurus yang mempunyai persamaan garis y = 3x – 6

Nilai Kognitif

60

Nilai Afektif

Nilai Psikomotorik


dan melalui titik (2,3) !

Paraf Guru

Paraf Orang Tua

BAB 4

SISTEM PERSAMAAN LINIER DUA VARIABEL Pembelajaran 4.1 Standar Kompetensi

= Memahami Sistem Persamaan linier Dua Variabel (SPLDV) dan Menggunakannya dalam pemecahan masalah.

Kompetensi Dasar

= Menyelesaikan Sistem Persamaan Linier Dua Variabel (SPLDV)

Indikator -

= 1. Membedakan PLDV dan SPLDV 2. Menyelesaikan Sistem Persamaan Linier Dua Variabel dengan menggunakan metode grafik

Tujuan Pembelajaran = Siswa dapat membedakan PLDV dan SPLDV Siswa dapat menyelesaikan SPLDV dengan menggunakan metode grafik

Persamaan Linear Satu Variabel Di Kelas VII, kamu telah mempelajari materi tentang persamaan linear satu variabel. Masih ingatkah kamu apa yang dimaksud dengan persamaan linear satu variabel? Coba kamu perhatikan bentuk-bentuk persamaan berikut. x+5=6 2r = 3 + 9 12 + y = 14 8p + 6 = 24 Bentuk-bentuk persamaan tersebut memiliki satu variabel seperti inilah yang dimaksud dengan linear satu variabel.

dan berpangkat satu. Bentuk persamaan

Persamaan Linier Dua Variabel (PLDV) Coba kamu perhatikan bentuk-bentuk persamaaan berikut. Sebutkan variabel dari masing-masing persamaan: 2x + 3y = 14

memiliki variabel…………dan………… masing-masing variabel berpangkat………

12m – n = 30


p + q + 3 = 10


12m – n = 30


r + 65 = 10


4a + 5b = b


Persamaan-persamaan tersebut memiliki dua variabel. Bentuk inilah yang dimaksud dengan Persamaan Linear Dua Variabel (PLDV).

BAB 4 SISTEM PERSAMAAN LINIER DUA VARIABEL

61


Jadi, Persamaan Linear Dua Variabel merupakan persamaan yang memiliki ............

variabel tunggal dan masing-masing berpangkat .............................

Cobalah tuliskan persamaan linear dua variabel yang lain! 1)

…………………………

2)

…………………………

3)

…………………………

Sistem Persamaan Linier Dua Variabel (SPLDV) Sistem persamaan linear dua variabel merupakan gabungan dua PLDV yang membentuk satu kesatuan (sistem) Misalkan : x+y=4 2x – y = 6 Atau bisa juga ditulis dalam bentuk : x + y = 4 dan 2x – y = 6 Cobalah tuliskan sistem persamaan linear dua variabel yang lain! 1) …………………………

2) …………………………

3) …………………………

…………………………

…………………………

…………………………

Berdasarkan jawaban kamu di atas, tuliskan secara umum Sistem Persamaan Linear Dua Variabel: .............. x + b .................. = p c .............. + ................... y = q

 Cobalah bedakan PLDV dengan SPLDV dan tulislah perbedaannya!

Penyelesaian: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Untuk menentukan himpunan penyelesaian dari system persamaan linear dua variabel dapat dilakukan dengan empat metode, yaitu metode grafik, metode substitusi, metode eliminasi,dan metode campuran (substitusi dan eliminasi).

Menyelesaikan SPLDV dengan Metode Grafik Penentuan himpunan penyelasaian SPLDV dengan metode grafik adalah sebagai berikut: 1) Gambarkan masing-masing garis yang dinyatakan oleh persamaan 2) Tentukan koordinat titik potong kedua garis yang merupakan penyelesaian dari sitem persamaan linear

62


“Lebih baik belajar satu halaman per hari daripada belajar satu buku tapi cuma sehari”

 Tentukan himpunan penyelesaian dari system persamaan 2x + y = 4 dan 2x + 3y = 8 dengan metode grafik.

Penyelesaian:  Gambarkan masing-masing persamaan:

2x + y = 4 tentukan titik potong kedua sumbu! (i) Jika memotong sumbu x, maka y = ....................... Sehingga 2x + ............. = 4 2x = ................ x = ................ Jadi titik potong dengan sumbu x adalah ( ............ , ............. ) (ii) Jika memotong sumbu y, maka x = ....................... Sehingga 2 ............. + y = 4 y = ............... Jadi titik potong dengan sumbu y adalah ( ............ , ............. ) (iii) Hubungkan kedua titik tersebut.

2x + 3y = 8 tentukan titik potong kedua sumbu! (i) Jika memotong sumbu x, maka y = ....................... Sehingga 2x + ............. = 4 2x = ................ x = ................ Jadi titik potong dengan sumbu x adalah ( ............ , ............. ) (ii) Jika memotong sumbu y, maka x = ....................... Sehingga 2 ........... + 3y = 8 y = ............... Jadi titik potong dengan sumbu y adalah ( ............ , ............. ) (iii) Hubungkan kedua titik tersebut.

kedua garis berpotongan di titik (……… , ………)

maka himpunan penyelesaiannya adalah (……… , ………)

Tentukanlah himpunan penyelesaian dari Sistem Persamaan Linier Dua Variabel di bawah ini dengan menggunakan metode grafik! 1) -x + 3y = -4 x - y = 10 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


63

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

2y =2x + 4 -y = x + 12 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

64



3)

4x – 9 = 6y 9x - 12 + 4y = 0 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


65



= Memahami Sistem Persamaan linier Dua Variabel (SPLDV) dan menggunakannya dalam pemecahan masalah.

Kompetensi Dasar


Indikator

= Menyelesaikan Sistem Persamaan Linier Dua Variabel dengan menggunakan metode substitusi

Tujuan Pembelajaran = Siswa dapat menyelesaikan SPLDV dengan menggunakan metode substitusi

Menyelesaikan SPLDV dengan Metode Substitusi Setelah kita belajar cara menentukan himpunan penyelesaian SPLDV menggunakan metode grafik, sekarang kita akan mempelajari cara menentukan himpunan penyelesaian SPLDV menggunakan metode substitusi. Langkah-langkah pengerjaan dengan menggunakan metode substitusi untuk mencari himpunan penyelesaian dari SPLDV adalah sebagai berikut. • • •

Ubahlah salah satu persamaan ke dalam bentuk x = .................. atau y = ................... Masukkan (substitusi) nilai x atau y yang diperoleh ke dalam persamaan yang kedua Nilai x atau y yang diperoleh kemudian disubstitusikan

ke dalam salah satu persamaan untuk memperoleh nilai variabel lainnya yang belum diketahui (x atau y).

 Tentukan himpunan penyelesaian dari SPLDV di bawah ini dengan metode substitusi 2x + y = 4 2x + 3y = 8

Penyelesaian: Langkah I pilih salah satu dari persamaan (yang sederhana).

paling

Langkah II Substitusikan pers. (3) ke pers. (2), maka diperoleh:

2x + y = 4 -------------------- pers. (1)

2x + 3(……… - …………) = 8

2x + 3y = 8 -------------------- pers. (2)

2x …………-…………… = 8

Ubah persamaan dalam bentuk y 2x + y = 4 y = .......... – ............ ------ pers. (3) Langkah III Substitusi nilai x ke persamaan (3), diperoleh y = ……………………… Jadi himpunan penyelesaian adalah (…………,…………)

66


(………… -………… )x = 8 - …………… x = …………………


Tentukanlah himpunan penyelesaian dari sistem persamaan linier dua variabel di bawah ini dengan menggunakan metode substitusi! 1)

a+b=4

2)

2a – b= 11

3)

3p + 4q -7 = 0 2p + q -3 = 0

...............................................................................................................

..................................................................................................................

...............................................................................................................

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...............................................................................................................

..................................................................................................................

...............................................................................................................

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a - 10 = 2b -2 = 4a + 6b ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ............................................................................................................... ...............................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


67



= Memahami Sistem Persamaan linier Dua Variabel (SPLDV) dan Menggunakannya dalam pemecahan masalah.

Kompetensi Dasar


Indikator

= Menyelesaikan Sistem Persamaan Linier Dua Variabel dengan menggunakan metode eliminasi dan campuran

Tujuan Pembelajaran = Siswa menyelesaikan Sistem Persamaan Linier Dua Variabel dengan menggunakan metode eliminasi dan campuran

Menyelesaikan SPLDV dengan metode Eliminasi Apakah kamu pernah mendengar kata eliminasi sebelumnya? Eliminasi artinya menghilangkan, maksudnya dengan cara menghilangkan salah satu variabel akan mendapatkan nilai dari variabel lainnya. Langkahlangkahnya adalah sebagai berikut: 1) Koefisien dari variabel yang akan dihilangkan harus sama 2) Jumlahkan atau kurangkan kedua persamaan yang diketahui agar koefisien dari variabel yang akan dihilangkan bernilai nol

 Tentukan himpunan penyelesaian dari SPLDV di bawah ini dengan metode eliminasi: 5x +2 y = - 4 x+y = 6

Penyelesaian:

5x +2 y = -4 x+y = 6

Langkah I Eliminasi variabel y maka koefisien dari y harus sama (untuk mencari nilai x) ........ x + 2 ...........

=

.............

× ....



............. x + .............y = .......

............. + y

=

.............

×2



.................. + 2 .......... = ....... ....... x + ....... = ....... ....... = x = ......

Langkah II Eliminasi variabel x maka koefisien dari x harus sama (untuk Mencari nilai y) ........ x + 2 ...........

=

.............

× ....



............. x + .............y = .......

............. + y

=

.............

×5

 .................. + ............... = ....... ........... + ....... y = ....... ....... = y = ......

Jadi himpunan penyelesaian dari sistem persamaan linier (.......... , ..........)

68



Tentukanlah himpunan penyelesaian dari sistem persamaan linier dua variabel di bawah ini dengan menggunakan metode eliminasi! 1)

2x + 3y = -6 x – 4y = -3 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Y = x+4 -y = 2x – 2 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

x = 3y - 7 y = 2x + 10 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

3x – 9 = y x–1+y=0 ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


69


Menyelesaikan SPLDV dengan metode Campuran Dalam pengerjaan soal persamaan linear dua variabel, terkadang kita menemukan kesulitan jika menggunakan metoda eliminasi untuk menentukan himpunan penyelesaiannya. Oleh karena itu, kita dapat menggunakan metode campuran, yaitu menentukan salah satu variabel x atau y dengan menggunakan metode eliminasi. Hasil yang diperoleh dari x atau y kemudian disubstitusikan ke salah satu persamaan linear dua variabel tersebut.

 Tentukan himpunan penyelesaian dari SPLDV menggunakan metode campuran! x + 2y = 7 dan 2x + 3y = 10

Penyelesaian : Mengeliminasi variabel x (untuk mencari y) ............ + 2y = 7

(× .......)

...........x + ......... = ..........

2x + ............. = 10

(× .......)

.............. + .............y = .......... y = .........

Substitusi y = ............... ke persamaan 2x + 3y = 10 2x + 3y = 10

⇔

2x + 3(..........) = 10

⇔

2x + ............ = ...............

⇔

..............x = ..............

⇔

x = .................

Jadi, himpunan penyelesaian dari sistem persamaan tersebut adalah {(........... , ..........)}.

Tentukan himpunan penyelesaian dari sistem persamaan berikut menggunakan metode campuran! 1)

x + y = 6; x,y ∈ R 3x – y = 10; x,y∈R

2) 2x + 4y = 6; x,y∈R 2x + 3y = 2; x,y∈R

3)

x + 2y = 3; x,y∈R 4x + 6y = 4; x,y∈R

4) 3x – 5y = 9; x,y∈R 4x – 7y = 13; x,y∈R

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

70


“Lebih baik belajar satu halaman per hari daripada belajar satu buku tapi cuma sehari” ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. 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Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


71



= Memahami sistem persamaan linear dua variabel (SPLDV) dan menggunakannya dalam pemecahan masalah.

Kompetensi Dasar

= Membuat model matematika dari masalah yang berkaitan dengan SPLDV.

Indikator

= Membuat model matematika dari masalah sehari-hari yang berkaitan dengan SPLDV.

Tujuan Pembelajaran = Siswa dapat membuat model matematika dari masalah sehari-hari yang berkaitan dengan SPLDV. Dalam kehidupan sehari-hari banyak masalah yang berkaitan dengan SPLDV. Untuk menyelesaikan masalah yang melibatkan dua variabel yang belum diketahui nilainya, kamu dapat menggunakan model matemtika, yaitu mengubah penulisan masalah dalam kalimat matematika.

 Ani dan Ina membeli buku dan pensil di toko yang sama. Ani membeli 4 buku dan 3 pensil dengan harga Rp 2.500,-. Ina membeli 2 buku dan 7 pensil dengan harga Rp 2.900,-. Buatlah pernyataan diatas kedalam model matematika!

Penyelesaiaan:

Misalkan: Satu buah buku = b Satu buah pensil = p Maka model matematikanya dapat ditulis sebagai berikut: 4 buku dan 3 pensil dengan harga Rp 2.500 model matematikanya: 4b + 3……… = 2.500 2 buku dan 7 pensil dengan harga Rp 2.900 model matematikanya: ………… + 7p = 2.900

Buatlah model matematika dari permasalahan berikut ini! 1)

Harga 4 ekor ayam dan 3 ekor itik Rp 55.000,-, sedangkan harga 3 ekor ayam dan 5 ekor itik Rp 47.500,................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Rina membeli 5 buku tulis dan 3 buku gambar seharga Rp 10.750,-. Budi membeli 7 buku tulis dan 5 buku gambar seharga Rp 16.250,-. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

72



3)

Jumlah dua bilangan cacah adalah 112 dan selisih kedua bilangan tersebut adalah 36. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Sebuah persegi panjang diketahui kelilingnya adalah 60 meter dan panjangnya adalah 6 meter lebihnya dari lebar. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

5)

Jumlah uang Andi ditambah 3 kali uang Budi adalah Rp 32.500,-. Sedangkan 2 kali uang Andi ditambah 4 kali uang Budi adalah Rp 50.000,................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


73



= Memahami sistem persamaan linear dua variabel (SPLDV) dan menggunakannya dalam pemecahan masalah.

Kompetensi Dasar

= Menyelesaikan model matematika dari masalah yang berkaitan dengan SPLDV dan penafsirannya.

Indikator

= Menyelesaikan model matematika dari masalah yang berkaitan dengan SPLDV dan penafsirannya.

Tujuan Pembelajaran = Siswa dapat menyelesaikan model matematika dari masalah yang berkaitan dengan SPLDV dan penafsirannya. Masih ingatkah kamu cara membuat model matematika dari masalah sehari-hari yang berkaitan dengan SPLDV?

 Didik membeli 3 buah buku tulis dan 4 buah pensil seharga Rp 4.400, sedangkan Bagus membeli 5 buah buku tulis dan 3 buah pensil seharga Rp 5.500. Tentukanlah harga buku tulis dan pensil tersebut dan berapa Rupiahkah yang harus dibayar oleh Robi jika ia membeli 6 buah buku tulis dan 2 buah pensil?

Penyelesaian:

Misalkan:

Buku tulis = x Pensil = y

Tentukanlah model matematikanya: ...................................

--------------- pers. (1)

...................................

--------------- pers. (2)

Tentukan penyelesaian dari persamaan diatas: Dari persamaan (1) dan (2) selesaikan dengan menggunakan metode campuran. 3x + ...........= 4.400

(× 5)

.......... + 3y = 5.500

(× 3)

........... + 20 y = ........... ............. +

9 y = ..........

11 y = ......... y= y = ......... Substitusi y = ........... ke persamaan (1) 3x + 4y = 4.400

⇔

3x + 4(..........)

= 4.400

⇔

3x + ................

= 4.400

⇔

3x = 4.400 -...........

⇔

3x = .................

⇔

x =

⇔

x = ………

Sehinga nilai x adalah ………………… dan nilai y adalah …………………

74


-


jadi, harga buku tulis = Rp ………………………… dan harga pensil = Rp ………………………………… Maka, harga 6 buah buku tulis dan 2 buah pensil

= 6 (……………) + 2 (……………) = (……………) + (……………) = ……………

1)

Susan membeli 5 buah apel dan 10 buah mangga dengan harga Rp 85.000, sedangkan Robert membeli 3 buah apel dan 5 buah mangga dengan harga Rp 46.000. Berapakah yang harus dibayar oleh Diki jika ia membeli 2 buah apel dan 7 buah mangga? ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Harga 4 buah penggaris dan 5 buah jangka adalah Rp 9.000. Harga 7 buah penggaris dan 3 buah jangka adalah Rp 7.500. Tentukanlah harga sebuah penggaris dan sebuah jangka. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Keliling suatu persegi panjang adalah 60 meter. Panjangnya adalah 6 meter lebihnya dari lebar. Tentukan panjang dan lebar persegi panjang tersebut! ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


75

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Banyak siswa putra dan putri adalah 48 anak. Siswa putra lebih banyak dari pada siswa putri. Selisih putra dan putri adalah 4 anak. Tentukan banyak masing-masing siswa! ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

5)

Umur Rudi 8 tahun lebih tua dari saudara perempuannya, Maria. Empat tahun yang lalu, tiga kali umur Maria sama dengan dua kali umur Rudi. Berapakah umur mereka sekarang? ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

76

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua

BAB 5

TEOREMA PYTHAGORAS Pembelajaran 5.1 Standar Kompetensi

= Menggunakan teorema pythagoras Dalam Pemecahan Masalah

Kompetensi Dasar

= Menggunakan Teorema Pythagoras untuk menentukan panjang sisi segitiga siku-siku

Indikator

= 1. 2. 3. 4.

Menentukan kuadrat suatu bilangan Menentukan akar kuadrat suatu bilangan Menghitung luas dan panjang sisi persegi Membuktikan teorema Pythagoras

Tujuan Pembelajaran = Siswa dapat membuktikan teorema pythagoras

Kuadrat dan akar kuadrat suatu bilangan Mari ingat kembali tentang kuadrat suatu bilangan! 1)

12

=

1

x

1

=

1

2)

3

2

=

........

x

........

=

........

3)

162

=

.......

.......

=

........

4)

( )

=

=

........

5)

0.52

=

=

........

x x

.......

x

.......

Masih ingatkah kamu cara menentukan akar kuadrat dari suatu bilangan? akar kuadrat dari bilangan 25 ditulis √ √ = 5, karena 52 = 25 Tentukan akar kuadrat dari bilangan berikut: 1)

√

= ………

karena ……… = ………

2)

√

= ………

karena ……… = ………

3)

√

= ………

karena ……… = ………

4)

√

= ………

karena ……… = ………

Luas dan panjang sisi persegi Ingatkah kamu pengertian persegi ? Persegi adalah bangun datar ......................................................................................................................................... .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

BAB 5 TEOREMA PYTHAGORAS

77


Untuk menentukan luas persegi dapat dilakukan dengan menghitung banyaknya persegi satuan yang membentuk persegi besar. Hitunglah luas setiap persegi dibawah ini dalam satuan!

1 satuan

4 satuan

............ satuan Gambar 1.

............ satuan

Hitunglah panjang sisi setiap persegi dibawah ini dalam satuan!

1 satuan

2 satuan

............ satuan Gambar 2.

Bandingkan luas (Gambar 1) dan panjang sisi persegi (Gambar 2), o jika luas 1 satuan,

maka panjang sisi

1 

1 =

12

o jika luas 4 satuan,

maka panjang sisi

2 

4 =

22

o jika luas ..... satuan,

maka panjang sisi

......



......

= ...... 2

o jika luas ..... satuan,

maka panjang sisi

......



......

= ...... 2

atau o jika panjang sisi 1 satuan, maka luas

1  1 =

√

o jika panjang sisi 2 satuan, maka luas

4  2 =

√

o jika panjang sisi ..... satuan, maka luas

.....

 ..... = √

o jika panjang sisi ..... satuan, maka luas

.....

 ..... = √

Apa yang dapat kamu simpulkan? luas persegi merupakan .............................. dari panjang sisi persegi. Panjang sisi persegi merupakan ........................ dari luas persegi. Dari kesimpulan di atas, jawablah pertanyaan dibawah ini

1)

78

Berapakah panjang sisi persegi jika diketahui luasnya sebagai berikut a)

100 satuan

= ......................................

b)

144 satuan

= ......................................


............ satuan

“Orang yang luar biasa itu sederhana dalam ucapan, tetapi hebat dalam tindakan”

2)

Berapakah luas persegi jika diketahui panjang sisinya sebagai berikut a)

9 satuan

= ......................................

b)

15 satuan

= ......................................

Menemukan Teorema Pythagoras Alat/ bahan : penggaris,gunting,lem

o Langkah Kerja 1)

Gambarkan sebuah segitiga siku-siku dengan ukuran sisi siku-sikunya 4 satuan dan 3 satuan

2)

Gambarkan persegi pada kedua sisi siku-siku yang berukuran 4 dan 3 satuan. hitunglah luas masing-masing persegi. Luas persegi pada sisi siku-siku I

= ...............................

satuan

Luas persegi pada sisi siku-siku II

= ...............................

satuan

3)

Ukurlah panjang sisi miring pada segitiga tersebut dengan menggunakan penggaris, kemudian gambarlah ukuran panjang sisi miring tersebut pada kertas kerja.

4)

Dari langkah (3) gambarlah persegi, kemudian gunting persegi tersebut.

5)

Tempelkan persegi yang telah diperoleh pada langkah (4) pada sisi miring segitiga di atas. Diperoleh luas persegi pada sisi miring = .............................. satuan Perhatikan luas persegi pada sisi miring dengan luas persegi pada sisi siku-siku I dan luas persegi pada sisi siku-siku II Luas persegi pada sisi miring

= luas persegi pada sisi siku-siku I +.................................

.................

= ..................................... + ..................................

Dari kegiatan diatas, dapat disimpulkan: Jadi, pada setiap segitiga siku-siku, ............................................................................................. ...............................................................................................................................................................

 Luas persegi pada sisi miring  Luas persegi pada sisi siku-siku I

= c2 2

=a

maka panjang sisi miring = c maka panjang sisi siku-siku I = .......


79


 Luas persegi pada sisi siku-siku II  Luas persegi pada sisi miring c

2

= b2

maka panjang sisi siku-siku II = .......

= luas persegi pada sisi siku-siku I + ................................ = ..................................... + ................................

Atau, Kuadrat sisi miring = .................................................................... + Kuadrat sisi siku-siku II Dari hasil kegiatan di atas, pembuktian teorema pythagoras dapat direpresentasikan sebagai berikut: c a

c 2 = a 2 + .................

Kesimpulan :

b2 = ............ - ...........

Kuadrat sisi miring sama dengan jumlah kuadrat sisi siku-sikunya

a2 = ........... - ............ b

1) Hitunglah: a)

62

= ...........................

c)

152

=

..............................

b)

( )

= ...........................

d)

2,42 =

..............................

c)

2,56 =

..............................

2) Carilah akar kuadrat dari:

3)

a)

144

= ...........................

b)

900

= ...........................

Berapakah panjang sisi persegi jika diketahui luasnya sebagai berikut a)

36 satuan

= ......................................

b)

64 satuan

= ......................................

c) 225 satuan = ..........................................

4) Berapakah luas persegi jika diketahui panjang sisinya sebagai berikut

80

a)

11 satuan

= ......................................

b)

20 satuan

= ......................................


c) 27 satuan = ..........................................


5) Gunakan teorema pythagoras untuk menyatakan persamaan-persamaan yang berlaku pada segitiga berikut:

..................................................................................................................................................................

a)

..................................................................................................................................................................

b

..................................................................................................................................................................

a

..................................................................................................................................................................

..................................................................................................................................................................

c

..................................................................................................................................................................

..................................................................................................................................................................

..................................................................................................................................................................

..................................................................................................................................................................

b)

e

..................................................................................................................................................................

f

..................................................................................................................................................................

..................................................................................................................................................................

..................................................................................................................................................................

d

..................................................................................................................................................................

..................................................................................................................................................................

i

..................................................................................................................................................................

c)

..................................................................................................................................................................

g

..................................................................................................................................................................

..................................................................................................................................................................

h

..................................................................................................................................................................

..................................................................................................................................................................

A

d)

..................................................................................................................................................................

..................................................................................................................................................................

..................................................................................................................................................................

C

..................................................................................................................................................................

..................................................................................................................................................................

B

Nilai Kognitif

..................................................................................................................................................................

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


81



= Menggunakan teorema pythagoras dalam pemecahan masalah

Kompetensi Dasar

= Menggunakan Teorema Pythagoras untuk menentukan panjang sisi segitiga

Indikator

= Menghitung panjang sisi segitiga siku-siku jika 2 sisi lain diketahui

Tujuan Pembelajaran = Siswa dapat menggunakan teorema pythagoras untuk siku-siku jika 2 sisi lainnya diketahui

siku-siku

menghitung panjang sisi segitiga

Pada pertemuan sebelumnya, kamu sudah menemukan teorema pythagoras, yaitu: Kuadrat sisi miring adalah

a

.......................................................................................................

c 2 = a 2 + .................

c

b Marilah kita menerapkan teorema pythagoras untuk menghitung panjang sisi segitiga siku-siku.

1.

Gunakanlah teorema pythagoras untuk menghitung nilai x.

Penyelesaian:

9

x

7 2.

x2

= 92 + ......................

x2

=

.......... + .................

x2

=

.....................

x

=

√

x

=

...............

Diketahui segitiga KLM dengan sudut siku-siku di K. Jika panjang sisi KL sebesar 7 cm, dan LM sebesar 25 cm, a. Gambarlah segitiga KLM b. Hitunglah panjang sisi KM.

Penyelesaian: b. KM2 =

a.

LM2 - ......................

KM2 =

……… - 72

KM2 =

…………… - ………………

KM2 =

…………………

KM =

√

KM = …………………

82



1)

Gunakanlah teorema pythagoras untuk menghitung nilai x.

.........................................................................................................................................................................

a)

.........................................................................................................................................................................

.........................................................................................................................................................................

x

8

.........................................................................................................................................................................

.........................................................................................................................................................................

.........................................................................................................................................................................

6

.........................................................................................................................................................................

b)

.........................................................................................................................................................................

12

.........................................................................................................................................................................

.........................................................................................................................................................................

9

.........................................................................................................................................................................

x

.........................................................................................................................................................................

.........................................................................................................................................................................

.........................................................................................................................................................................

26

c)

.........................................................................................................................................................................

x

10

.........................................................................................................................................................................

.........................................................................................................................................................................

.........................................................................................................................................................................

2)

Hitunglah nilai y pada setiap segitiga berikut ini.

....................................................................................................................................................................................

a)

....................................................................................................................................................................................

8

y

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

y

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

b)

....................................................................................................................................................................................

20

4y

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

3y

....................................................................................................................................................................................

....................................................................................................................................................................................

....................................................................................................................................................................................

3) Diketahui segitiga PQR dengan sudut siku-siku di P. Jika panjang sisi PQ sebesar 12 cm, dan QR sebesar 13 cm, a) Gambarlah segitiga tersebut.

b) Tentukan panjang sisi PR


83

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Panjang hipotenusa suatu segitiga siku-siku adalah 15 cm, sedangkan panjang sisi siku-sikunya adalah sebesar 12 cm dan x cm. Berapakah nilai x? ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

5)

Pada gambar di bawah, diketahui panjang sisi AB sebesar 12 cm, sisi BC sebesar 25 cm. Tentukan panjang sisi AD.

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

84

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= 1. menentukan jenis-jenis segitiga jika panjang ketiga sisinya diketahui. 2. menentukan triple pythagoras.

Tujuan Pembelajaran = 1. siswa dapat menentukan jenis-jenis segitiga jika panjang ketiga sisinya diketahui. 2. siswa dapat menentukan triple pythagoras.

Jenis Segitiga jenis-jenis segitiga ditinjau dari besar sudutnya yang telah dipelajari yaitu, 1)

Segitiga siku-siku adalah, segitiga yang besar salah satu sudutnya .....................................

2)

Segitiga ............................

adalah, segitiga yang salah satu sudutnya  900.

3)

Segitiga .............................

adalah, segitiga .....................................................................

Untuk menentukan jenis segitiga dari kebalikan teorema pythagoras, kerjakanlah kegiatan berikut Alat/bahan : Lidi, isolasi.

o Kegiatan 1 1)

Ambillah 3 buah lidi yang panjangnya masing-masing sebesar 6 cm, 8 cm, dan 10 cm.

2)

Bentuklah sebuah segitiga dan tempelkan pada LKS ditempat yang telah tersedia.


85


3)

Jenis segitiga apakah yang terbentuk? Yaitu: segitiga .....................................................

4) Misalkan sisi terpanjang = c

maka nilai c = ..................

dan c2 = .........................

sisi kedua

=b

maka nilai b = ..................

dan b2 = .........................

sisi ketiga

=a

maka nilai a = ..................

dan a2 = .........................

a2 + b2 = .............. +.................. = ................... Maka, c2

..........

a2 + b2 (isilah tanda  ,  , atau )

Jenis segitiga apa yang dapat kamu simpulkan dari hasil kegiatan di atas? Jenis segitiga dari hasil kegiatan di atas adalah ..........................................., karena memiliki hubungan

c2 .......... a2 + b2

o Kegiatan 2 1)


2) 3)

Bentuklah sebuah segitiga dan tempelkan pada LKS ditempat yang telah tersedia. Jenis segitiga apakah yang terbentuk? Yaitu: segitiga .....................................................


maka nilai c = ..................

dan c2 = .........................

sisi kedua

=b

maka nilai b = ..................

dan b2 = .........................

sisi ketiga

=a

maka nilai a = ..................

dan a2 = .........................

a2 + b2 = .............. +.................. = ...................

86



Maka, c2

..........

a2 + b2 (isilah tanda  ,  , atau  )


c2 .......... a2 + b2

o Kegiatan 3 1)


2)

Bentuklah sebuah segitiga dan tempelkan pada LKS ditempat yang telah tersedia.

3)

Jenis segitiga apakah yang terbentuk? Yaitu: segitiga .....................................................


maka nilai c = ..................

dan c2 = .........................

sisi kedua

=b

maka nilai b = ..................

dan b2 = .........................

sisi ketiga

=a

maka nilai a = ..................

dan a2 = .........................

a2 + b2 = .............. +.................. = ................... Maka, c2

..........

a2 + b2 (isilah tanda  ,  , atau  )


c2 .......... a2 + b2


87


Triple Pythagoras Dengan memperhatikan kuadrat sisi terpanjang dan jumlah kuadrat 2 sisi lain, kamu dapat menentukan jenis segitiga.

Perhatikan pasangan dari 3 bilangan berikut! Manakah yang membentuk segitiga siku-siku? 1)

6 , 8 , 10

Penyelesaian: 102 2

6 +8

= 2

................................

=

......................... + ............................ = ...................................

Karena, 10

2

............. 62 + 82

pasangan bilangan 6, 8, 10 membentuk jenis segitiga ................................. dan disebut triple pythagoras. 2) 3 , 5 , 6

Penyelesaian: 62

=

..................................

.............. + .............. Karena, 6

2

= .............................. + .............................. = ............................

............. 32 + 52

pasangan bilangan 3, 5, 6 tidak membentuk segitiga siku-siku, dan bukan tripel pythagoras

Kesimpulan: Tripel Pytagoras adalah tiga bilangan asli ...................................................................................................... ...................................................................................................................................... ...................................................................................................................................... Sebutkanlah 2 pasang triple pythagoras yang lain: 1) …………………………………… karena

…………… = …………… + ……………

2) …………………………………… karena

…………… = …………… + ……………

1) Tentukan jenis segitiga jika diketahui panjang sisi-sisinya sebagai berikut: a)

88

9 cm, 12 cm, 15 cm

b)

5 cm, 8 cm, 12 cm

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................

c)

d)

9 cm, 13 cm, 17 cm

8 cm, 15 cm, 20 cm

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................

....................................................................................................

..............................................................................................................



e)

7 cm, 24 cm, 25 cm

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2) Tentukan apakah pasangan-pasangan bilangan berikut termasuk triple pythagoras atau bukan. a)

12, 16, 20

b)

7, 8, 11

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

c)

d)

6, 8, 10

5, 3, 2

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

....................................................................................................

...............................................................................................................

e)

8, 15, 17

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3) Pada segitiga ABC diketahui panjang AB=10 cm, BC=24 cm, dan AC=26 cm. a) Tunjukkan bahwa segitiga ABC tersebut segitiga siku-siku. b) Tentukan letak sudut siku-siku dari segitiga ABC tersebut? ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

Nilai Afektif

Nilai Psikomotorik

Paraf Guru

Paraf Orang Tua


89




Kompetensi Dasar

= Memecahkan masalah pada bangun datar yang berkaitan dengan teorema pythagoras

Indikator

= Menghitung perbandingan sisi-sisi segitiga siku-siku istimewa (salah satu sudutnya 300, 450, 600)

Tujuan Pembelajaran = Siswa dapat menentukan perbandingan sisi-sisi segitiga siku-siku istimewa.

Segitiga dengan sudut 450 o Perbandingan sisi-sisi segitiga siku-siku dengan sudut 450 Berikut Langkah kerja untuk menemukan perbandingan segitiga yang sudutnya 450: 1) Gambarlah sebuah persegi ABCD pada tempat yang tersedia 2) Misalkan panjang sisi persegi adalah a 3) Hubungkan garis diagonal AC 4) Ada berapa buah segitiga yang terbentuk? Sebutkanlah nama segitiga yang terbentuk! segitiga ........................................ dan segitiga ......................................... 5) Berapakah besar sudut BAC yang terbentuk?  BAC = ................................. 6) Gambarkan kembali segitiga ABC yang terbentuk dari langkah (4) dan tulis besar sudut-sudut yang terbentuk. Gambar:

Dengan menggunakan teorema pythagoras tentukan panjang sisi AC.

90

AC2

=

AB2 + .......................

AC2

=

a2 + .......................

AC2

=

.......................

AC

=

√

AC

=

a√



Dengan demikian perbandingan panjang sisi segitiga siku-siku ABC adalah: AB : BC : AC =

a : .............. : a√

AB : BC : AC = ………… :

1

: ................

Diketahui segitiga ABC dengan panjang sisi AB adalah 7 cm, tentukan panjang sisi AC seperti pada gambar di bawah ini!

Penyelesaian:

AB : AC = 1 : ………….. 7 : ……..=……… : ……… ……………. =…………………. AC =………..

Segitiga dengan sudut 600 dan 300 Untuk menemukan perbandingan segitiga yang sudutnya 600 ikuti langkah kerja berikut ini. 1) Gambarlah sebuah segitiga sama sisi ABC pada tempat yang tersedia,dengan panjang sisi 2a 2) Kemudian bagi dua segi tiga tersebut dengan menarik garis tinggi CD 3) Ada berapa buah segitiga siku-siku yang terbentuk? Sebutkanlah nama segitiga yang terbentuk! 4) 5)

segitiga ........................................ dan segitiga ......................................... Berapakah besar sudut BAC yang terbentuk?  DBC = ................................. Gambarkan kembali segitiga DBC yang terbentuk dari langkah 4) dan tulis besar sudut-sudut yang terbentuk.

Dengan menggunakan Teorema Pythagoras tentukan panjang BC CD2 = ............ - ............ CD2 = ............ - ............ CD2 = ............ CD2 = ............ Dengan demikian perbandingan panjang sisi segitiga siku-siku DBC adalah: BD : CD : BC = ............ : ............ : 2a BD : CD : BC = ............ : ............ : ............


91


1)

Hitunglah panjang sisi yang belum diketahui. a) .................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

b)

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

c)

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................

2)

Diketahui segitiga ABC siku-siku di C dan besar sudut BAC = 300 jika panjang AB= 8cm. Hitunglah panjang AC dan BC. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Diketahui segitiga PQR siku-siku di Q dengan panjang sisi PR adalah QPR=450, tentukan panjang PQ dan QR.

√ cm. Jika besar sudut

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

92

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar

= Memecahkan masalah pada bangun datar yang berkaitan dengan teorema pythagoras

Indikator

= Menghitung panjang diagonal pada bangun datar dengan menggunakan teorema pythagoras.

Tujuan Pembelajaran = Siswa dapat menerapkan teorema pythagoras pada bangun datar. Pada kondisi tertentu, teorema pythagoras dapat digunakan dalam perhitungan bangun datar.

Sebuah persegi ABCD mempunyai panjang sisi 8 cm, tentukanlah: a) Sketsa gambarnya. b) Tentukan panjang diagonal.

Penyelesaian:

a.

b. Salah satu diagonalnya AC AC2 = .......... + .......... AC2 = ..................... AC = ....................... AC =........................

1)

Perhatikan gambar persegi panjang ABCD dibawah ini. Jika diketahui ukuran panjang dan lebar persegi panjang bertuurut-turut adalah 15 cm dan 8 cm, maka tentukanlah: a) Luas persegi panjang ABCD b) Panjang diagonal BD.

Penyelesaian: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Perhatikan trapesium ABCD pada gambar disamping. Diketahui panjang alas trapesium adalah 7 cm, panjang sisi adalah 4 cm dan tinggi trapesium adalah 4 cm. tentukanlah: a) Panjang sisi miring AD. b) Keliling trapesium ABCD. c) Luas Trapesium ABCD.


93


Penyelesaian: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Gambar disamping adalah bangun datar layang-layang PQRS. Jika diketahui panjang QS=52 cm, maka tentukanlah: a) Panjang ST b) Panjang PQ c) Luas layang-layang PQRS

Penyelesaian: ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

94

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua




Kompetensi Dasar


Indikator

= Menerapkan teorema pythagoras dalam pemecahan masalah

Tujuan Pembelajaran = Siswa dapat menerapkan teorema pythagoras dalam pemecahan masalah Dalam kehidupan sehari-hari banyak sekali masalah-masalah yang dapat dipecahkan menggunakan teorema Pythagoras. Untuk mempermudah perhitungan, alangkah baiknya jika permasalahan tersebut direpresentasikan dalam bentuk gambar. Coba kamu perhatikan dan pelajari contoh-contoh soal berikut!

Sebuah tangga bersandar pada tembok. Jarak antara kaki tangga dengan tembok 2 meter dan jarak antara tanah dan ujung atas tangga 8 meter. Hitunglah panjang tangga ! 1) Representasikan permasalahan di atas dan gambarkanlah. 2) Gunakan teorema Pythagoras untuk menghitung panjang sisi BC. ................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

1)

Sebuah televisi memiliki lebar layar 15 cm dan tinggi layar 8 cm. Tentukanlah: a) panjang diagonal layar televisi tersebut. b) keliling layar televisi tersebut. c) luas layar televisi tersebut. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

2)

Dua buah tiang berdampingan berjarak 24 m. Jika tinggi tiang masing-masing adalah 22 m dan 12 m, hitunglah panjang kawat penghubung antara ujung tiang tersebut.


95

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

3)

Sebidang sawah berbentuk persegi panjang berukuran 40 m x 9 m. Sepanjang keliling dan kedua diagonalnya akan dibuat pagar dengan biaya Rp 25.000 per meter. Hitunglah: a) panjang pagar. b) biaya pembuatan pagar. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

4)

Amat akan menanam pohon di sekeliling kebunnya yang berbentuk seperti gambar disamping. Jarak antara pohon yang satu dengan yang lain adalah 2 m.

a) Tentukan panjang AB b) Tentukan keliling Kebun c) Berapa banyak pohon yang dapat di tanam?

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

96

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua



= Menggunakan teorema pythagoras dalam pemecahan masalah

Kompetensi Dasar


Indikator

= Menghitung panjang diagonal ruang dan diagonal bidang pada bangun ruang

Tujuan Pembelajaran = Siswa dapat menghitung panjang diagonal ruang dan diagonal bidang pada bangun ruang Selain dimanfaatkan pada segitiga siku-siku, teorema Pythagoras juga dapat digunakan pada bangun ruang untuk mencari panjang diagonal bidang dan diagonal ruang. Hal ini dikarenakan diagonal bidang dan diagonal ruang merupakan sisi miring bagi sisi bidangnya.

Diketahui kubus ABCD.EFGH dengan panjang AB= 15 cm. Hitunglah: a) Panjang diagonal bidang AC. b) Panjang Diagnal ruang AG.

Penyelesaian Kubus merupakan bangun ruang yang semua sisinya sama panjang, oleh karena itu panjang AB = 15 cm dan Panjang BC=15 cm.

1)

Pada limas T.PQRS di samping, alas limas berbentuk persegi dengan panjang sisi8 cm, sedangkan panjang TO =12 cm a) Hitunglah panjang TU b) Hitung luas Δ TQR

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................


97

MGMP Matematika MATRIKS Wil. Timur Kota Langsa, Aceh ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Pada gambar disamping balok ABCD.EFGH dengan sisi alas ABCD dan sisi atas EFGH. Panjang rusuk AB=8 cm, BC=6 cm dan CG=4 cm

2)

a) Hitung panjang AC b) Hitung luas bidang ACGE c) Hitung keliling bidang ACGE

................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................

Nilai Kognitif

98

Nilai Afektif

Nilai Psikomotorik


Paraf Guru

Paraf Orang Tua


TETA-TEKI SILANG

1

2

3

5

4

6

7

8

9

10

11

12 13

14

15

16 18

20

17

19 21

22

Mendatar:

Menurun:

1.

1.

Kuadrat 35

2.

2. 3.

Kuadrat 7 14

2

4.

√

5.

√

 50 √

2

5.

Kuadrat 34

6.

9 √

7.

502

9.

Triple pytagoras: 20, ....., 12

8.

Luas persegi dengan panjang 40 satuan (balik)

11.

262  252

52  22

13.

√

9. 10.

21  √

14.

112

15.

132  52

12.

Triple pytagoras: ....., 15, 8

172

15.

82  22

17.

16.

612  332

18.

502  92

20. ABC siku-siku di A. Jika AB=240 cm dan AC=320 cm, maka BC= ...... 21.

√

22. √

19.

Nilai ab = .........

√


99

DAFTAR PUSTAKA

Dewi Nuharini, 2008, Matematika Konsep dan Aplikasinya: untuk SMP Kelas VIII, Buku Sekolah Elektronik (BSE), Pusat Pembukuan Departemen Pendidikan Nasional, Jakarta. Endah Budi Rahayu, 2008, Contextual Teaching and Learning Matematika: Sekolah Menengah Pertama/Madrasah Tsanawiyah Kelas VIII Edisi 4, Buku Sekolah Elektronik (BSE), Pusat Pembukuan Departemen Pendidikan Nasional, Jakarta. Heru Nugroho, 2009, Matematika 2: SMP dan MTs Kelas VIII, Buku Sekolah Elektronik (BSE), Pusat Pembukuan Departemen Pendidikan Nasional, Jakarta. Nuniek Avianti Agus, 2008, Mudah Belajar Matematika untuk kelas VIII Sekolah Menengah Pertama/Madrasah Tsanawiyah, Buku Sekolah Elektronik (BSE), Pusat Pembukuan Departemen Pendidikan Nasional, Jakarta. Syamsul Junaidi, dkk, 2004, Matematika SMP untuk Kelas VIII, Penerbit Erlangga (Esis), Jakarta.

100 LKS MATEMATIKA KELAS VIII SEMESTER 1

TIM PENYUSUN

No

Nama

Instansi

Keterangan

Universitas Zawiyah Cot Kala, Langsa

Pembimbing

1

Yenny Suzana, M.Pd.

2

Hardani, S.Pd.

SMPN 1 LANGSA

Ketua

3

Intan Yuliani, S.Pd.

SMPN 4 LANGSA

Sekretaris

4

Muhammad Yusuf, S.Pd

SMPN 1 LANGSA

Bendahara

5

Nur Asni, S.Pd

SMPN 1 LANGSA

Anggota

6

Ida, S.Pd

SMPN 1 LANGSA

Anggota

7

Effendi

SMPN 1 LANGSA

Anggota

8

Meltda Silvira, S.Pd.

SMPN 1 LANGSA

Anggota

9

Elmayenti, S.Pd.

SMPN 1 LANGSA

Anggota

10

Nuraini, S.Pd.

SMPN 5 LANGSA

Anggota

11

Syah Misriah, S.Pd.I.

SMPN 5 LANGSA

Anggota

12

Zuraidah

SMPN 5 LANGSA

Anggota

13

Taufik Zulhidayat, S.Pd.

SMPN 7 LANGSA

Anggota

14

Ibrahim, S.Pd.

SMPN 7 LANGSA

Anggota

15

Sarmila, S.Pd.

SMPN 10 LANGSA

Anggota

16

Widiya Mandasari, S.Pd.

SMPN 10 LANGSA

Anggota

17

Muhammad Irwansa, S.Pd.

SMPN 10 LANGSA

Anggota

18

Listriana, S.Pd.

SMPN 12 LANGSA

Anggota

19

Epi Yulida, S.Pd.I.

SMPN 12 LANGSA

Anggota

20

Iswadi, S.Pd.

SMPN 13 LANGSA

Anggota

21

Suyani, S.Pd.

SMPN 13 LANGSA

Anggota

22

Salpia, S.T.

SMPN 13 LANGSA

Anggota

2) Drs. Mustafa, M.Pd., selaku Kepala Dinas Pendidikan Kota Langsa

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