Roboguru

Diketahui segitiga  dengan titik sudut masing-masing ,  dan . Tentukan: d. Gambar segitiga  dan bayangan segitiga  pada rotasi  berlawanan arah jarum jam dengan pusat rotasi .

Pertanyaan

Diketahui segitiga ABC dengan titik sudut masing-masing straight A open parentheses 2 comma space 1 close parenthesesstraight B open parentheses 4 comma space 2 close parentheses dan straight C open parentheses 2 comma space 4 close parentheses. Tentukan:

d. Gambar segitiga ABC dan bayangan segitiga ABC pada rotasi 60 degree berlawanan arah jarum jam dengan pusat rotasi straight P open parentheses 1 comma space 1 close parentheses.

Pembahasan Soal:

Rotasi terhadap titik asal straight P open parentheses h comma space k close parentheses, dengan sudut rotasi theta, dapat dihitung dengan rumus berikut.

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos space theta end cell cell negative sin space theta end cell row cell sin space theta end cell cell cos space theta end cell end table close parentheses open parentheses table row cell x minus h end cell row cell y minus k end cell end table close parentheses plus open parentheses table row h row k end table close parentheses

Diketahui segitiga ABC dengan titik sudut masing-masing straight A open parentheses 2 comma space 1 close parenthesesstraight B open parentheses 4 comma space 2 close parentheses dan straight C open parentheses 2 comma space 4 close parentheses.

Akan ditentukan gambar segitiga ABC dan bayangan segitiga ABC pada rotasi 60 degree berlawanan arah jarum jam dengan pusat rotasi straight P open parentheses 1 comma space 1 close parentheses. Karena rotasi berlawanan arah jarum jam, maka Rotasi bernilai positif.

Menentukan bayangan titik straight A open parentheses 2 comma space 1 close parentheses.

begin mathsize 12px style straight A open parentheses 2 comma space 1 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight A apostrophe open parentheses x apostrophe comma space y apostrophe close parentheses table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space open parentheses 60 degree close parentheses end cell cell negative sin space open parentheses 60 degree close parentheses end cell row cell sin space open parentheses 60 degree close parentheses end cell cell cos space open parentheses 60 degree close parentheses end cell end table close parentheses open parentheses table row cell 2 minus 1 end cell row cell 1 minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half end cell cell negative 1 half square root of 3 end cell row cell 1 half square root of 3 end cell cell 1 half end cell end table close parentheses open parentheses table row 1 row 0 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half plus 0 end cell row cell 1 half square root of 3 plus 0 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half end cell row cell 1 half square root of 3 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 1 half end cell row cell 1 half square root of 3 plus 1 end cell end table close parentheses end cell end table straight A open parentheses 2 comma space 1 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight A apostrophe open parentheses 1 1 half comma space 1 half square root of 3 plus 1 close parentheses end style 

Menentukan bayangan titik straight B open parentheses 4 comma space 2 close parentheses.

begin mathsize 12px style straight B open parentheses 4 comma space 2 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight B apostrophe open parentheses x apostrophe comma space y apostrophe close parentheses table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space open parentheses 60 degree close parentheses end cell cell negative sin space open parentheses 60 degree close parentheses end cell row cell sin space open parentheses 60 degree close parentheses end cell cell cos space open parentheses 60 degree close parentheses end cell end table close parentheses open parentheses table row cell 4 minus 1 end cell row cell 2 minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half end cell cell negative 1 half square root of 3 end cell row cell 1 half square root of 3 end cell cell 1 half end cell end table close parentheses open parentheses table row 3 row 1 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 3 over 2 minus 1 half square root of 3 end cell row cell 3 over 2 square root of 3 plus 1 half end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 1 half minus 1 half square root of 3 end cell row cell 1 1 half square root of 3 plus 1 half end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 2 1 half minus 1 half square root of 3 end cell row cell 1 1 half square root of 3 plus 1 1 half end cell end table close parentheses end cell end table straight B open parentheses 4 comma space 2 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight B apostrophe open parentheses 2 1 half minus 1 half square root of 3 comma space 1 1 half square root of 3 plus 1 1 half close parentheses end style

Menentukan bayangan titik straight C open parentheses 2 comma space 4 close parentheses.

begin mathsize 12px style straight C open parentheses 2 comma space 4 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight C apostrophe open parentheses x apostrophe comma space y apostrophe close parentheses table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space open parentheses 60 degree close parentheses end cell cell negative sin space open parentheses 60 degree close parentheses end cell row cell sin space open parentheses 60 degree close parentheses end cell cell cos space open parentheses 60 degree close parentheses end cell end table close parentheses open parentheses table row cell 2 minus 1 end cell row cell 4 minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half end cell cell negative 1 half square root of 3 end cell row cell 1 half square root of 3 end cell cell 1 half end cell end table close parentheses open parentheses table row 1 row 3 end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half minus 3 over 2 square root of 3 end cell row cell 1 half square root of 3 plus 3 over 2 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half minus 1 1 half square root of 3 end cell row cell 1 half square root of 3 plus 1 1 half end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 1 half minus 1 1 half square root of 3 end cell row cell 1 half square root of 3 plus 2 1 half end cell end table close parentheses end cell end table straight C open parentheses 2 comma space 4 close parentheses rightwards arrow with open square brackets straight P open parentheses 1 comma space 1 close parentheses comma space straight R open parentheses 60 degree close parentheses close square brackets on top straight C apostrophe open parentheses 1 1 half minus 1 1 half square root of 3 comma space 1 half square root of 3 plus 2 1 half close parentheses end style

Jadi, diperoleh titik koordinat bayangan segitiga ABC adalah straight A apostrophe open parentheses 1 1 half comma space 1 half square root of 3 plus 1 close parenthesesstraight B apostrophe open parentheses 2 1 half minus 1 half square root of 3 comma space 1 size 12px 1 over size 12px 2 square root of size 12px 3 size 12px plus size 12px 1 size 12px 1 over size 12px 2 close parentheses dan straight C apostrophe open parentheses 1 1 half minus 1 1 half square root of 3 comma space size 12px 1 over size 12px 2 square root of size 12px 3 size 12px plus size 12px 2 size 12px 1 over size 12px 2 close parentheses .

Jika digambarkan akan menjadi seperti berikut.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

Y. Fathoni

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Terakhir diupdate 01 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Tentukan persamaan bayangan dari setiap rotasi berikut: c. lingkaran  terhadap  sejauh  berlawanan arah jarum jam.

Pembahasan Soal:

Rotasi terhadap  left parenthesis 4 comma 5 right parenthesis sejauh begin mathsize 14px style straight pi end style berlawanan jarum jam maka,

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space 60 end cell cell negative sin space 60 end cell row cell sin space 60 end cell cell cos space 60 end cell end table close parentheses open parentheses table row cell x minus 4 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half end cell cell negative 1 half square root of 3 end cell row cell 1 half square root of 3 end cell cell 1 half end cell end table close parentheses open parentheses table row cell x minus 4 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half open parentheses x minus 4 close parentheses minus 1 half square root of 3 left parenthesis y minus 5 right parenthesis end cell row cell 1 half square root of 3 left parenthesis x minus 4 right parenthesis plus 1 half left parenthesis y minus 5 right parenthesis end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 2 minus 1 half square root of 3 y minus 5 over 2 end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y minus 5 over 2 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 1 half square root of 3 y minus 2 plus 4 minus 5 over 2 end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y minus 5 over 2 plus 5 end cell end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 1 half square root of 3 y minus 1 half end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y plus 5 over 2 end cell end table close parentheses end cell row cell x apostrophe end cell equals cell 1 half x minus 1 half square root of 3 y minus 1 half end cell row cell 1 half x end cell equals cell x apostrophe plus 1 half square root of 3 y plus 1 half end cell row x equals cell 2 x apostrophe plus square root of 3 y plus 1 end cell row blank blank blank row cell y apostrophe end cell equals cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y plus 5 over 2 end cell row cell 1 half y end cell equals cell y apostrophe minus 1 half square root of 3 x plus 2 square root of 3 minus 5 over 2 end cell row y equals cell 2 y apostrophe minus square root of 3 x plus 4 square root of 3 minus 5 end cell row blank blank blank end table

Maka di dapat table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 5 end table. Cari dulu pusat lingkaran L identical to x squared plus y squared plus 2 x minus 2 y minus 5 equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus y squared plus 2 x minus 2 y minus 5 end cell equals 0 row cell x squared plus y squared plus A x plus B y plus C end cell equals 0 row P equals cell open parentheses negative A over 2 comma negative B over 2 close parentheses end cell row P equals cell open parentheses negative 2 over 2 comma negative fraction numerator left parenthesis negative 2 right parenthesis over denominator 2 end fraction close parentheses end cell row P equals cell open parentheses negative 1 comma 1 close parentheses end cell end table

Di dapatkan x equals negative 1 dan y equals 1 kemudian masukkan ke persamaan,

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell 2 x apostrophe plus square root of 3 y plus 1 end cell row cell 2 x apostrophe end cell equals cell x minus square root of 3 y minus 1 end cell row cell x apostrophe end cell equals cell fraction numerator x minus square root of 3 y minus 1 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell fraction numerator negative 1 minus square root of 3 space left parenthesis 1 right parenthesis minus 1 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell fraction numerator negative 2 over denominator 2 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell negative 1 minus 1 half square root of 3 end cell row blank blank blank row y equals cell 2 y apostrophe minus square root of 3 x plus 4 square root of 3 minus 5 end cell row cell 2 y apostrophe end cell equals cell y plus square root of 3 x minus 4 square root of 3 plus 5 end cell row cell y apostrophe end cell equals cell fraction numerator y plus square root of 3 x minus 4 square root of 3 plus 5 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 1 plus square root of 3 left parenthesis negative 1 right parenthesis minus 4 square root of 3 plus 5 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 6 minus square root of 3 minus 4 square root of 3 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 6 minus 5 square root of 3 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell 3 minus 5 over 2 square root of 3 end cell end table

Kemudian cari jari-jari,

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of open parentheses fraction numerator negative A over denominator 4 end fraction close parentheses squared plus open parentheses fraction numerator negative B over denominator 4 end fraction close parentheses squared minus C end root end cell equals cell square root of open parentheses fraction numerator negative 2 over denominator 4 end fraction close parentheses squared plus open parentheses fraction numerator negative left parenthesis negative 2 right parenthesis over denominator 4 end fraction close parentheses squared minus left parenthesis negative 5 right parenthesis end root end cell row blank equals cell square root of open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses squared plus open parentheses 1 half close parentheses squared plus 5 end root end cell row blank equals cell square root of 1 fourth plus 1 fourth plus 5 end root end cell row blank equals cell square root of 2 over 4 plus 20 over 4 end root end cell row blank equals cell square root of 22 over 4 end root end cell row blank equals cell 1 half square root of 22 end cell end table

Maka di dapatkan, table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 5 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table. Masukkan ke persamaan lingkaran,

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x minus a right parenthesis squared plus left parenthesis x minus b right parenthesis squared end cell equals cell r squared space end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell open parentheses 1 half square root of 22 close parentheses squared end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 1 fourth times 22 end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 22 over 4 end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 5 comma 5 end cell end table
 

Jadi, bayangan persamaan garisnya adalah Error converting from MathML to accessible text..

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Roboguru

Diketahui suatu garis l: 12x + 5y = 36 dirotasikan sejauh α berlawanan arah jarum jam dengan pusat (3,5) sehingga diperoleh garis k. Jika  dengan interval 270° < α < 360°. Maka ordinat titik pot...

Pembahasan Soal:

Perhatikan bahwa begin mathsize 14px style cos invisible function application alpha equals 12 over 13 end style dengan interval 270° < α < 360°.
Karena 270° < α < 360°, maka α berada di kuadran IV.
Misalkan β = 360° - α, maka β akan berada di kuadran I.
Didapat bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos invisible function application beta end cell equals cell cos invisible function application open parentheses 360 degree minus alpha close parentheses end cell row cell cos invisible function application beta end cell equals cell cos invisible function application alpha end cell row cell cos invisible function application beta end cell equals cell 12 over 13 end cell end table end style

Karena begin mathsize 14px style cos invisible function application alpha equals 12 over 13 end style, maka didapat gambar segitiga siku-siku berikut.

Dengan menggunakan Teorema Pythagoras akan didapat x = 5. Sehingga didapat begin mathsize 14px style sin invisible function application beta equals x over 13 equals 5 over 13 end style.
Perhatikan bahwa

begin mathsize 14px style sin invisible function application beta equals sin invisible function application open parentheses 360 degree minus alpha close parentheses equals negative sin invisible function application alpha end style

Sehingga

begin mathsize 14px style sin invisible function application alpha equals negative sin invisible function application beta equals negative 5 over 13 end style

Selanjutnya misalkan titik A(x,y) terletak pada garis l.
Kemudian titik tersebut juga dirotasikan terhadap pusat (3,5). Sehingga a = 3 dan b = 5.
Besar sudut rotasinya adalah α berlawanan arah jarum jam. 
Misalkan bayangannya adalah A’(x’,y’), maka didapatkan hubungan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x to the power of apostrophe end cell row cell y to the power of apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos invisible function application alpha end cell cell negative sin invisible function application alpha end cell row cell sin invisible function application alpha end cell cell cos invisible function application alpha end cell end table close parentheses open parentheses table row cell x minus a end cell row cell y minus b end cell end table close parentheses plus open parentheses table row a row b end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe end cell row cell y to the power of apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 12 over 13 end cell cell negative open parentheses negative 5 over 13 close parentheses end cell row cell negative 5 over 13 end cell cell 12 over 13 end cell end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 3 row 5 end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe end cell row cell y to the power of apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 12 over 13 end cell cell 5 over 13 end cell row cell negative 5 over 13 end cell cell 12 over 13 end cell end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 3 row 5 end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe end cell row cell y to the power of apostrophe end cell end table close parentheses end cell equals cell 1 over 13 open parentheses table row 12 5 row cell negative 5 end cell 12 end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 3 row 5 end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe end cell row cell y to the power of apostrophe end cell end table close parentheses minus open parentheses table row 3 row 5 end table close parentheses end cell equals cell 1 over 13 open parentheses table row 12 5 row cell negative 5 end cell 12 end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell 1 over 13 open parentheses table row 12 5 row cell negative 5 end cell 12 end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 13 open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row 12 5 row cell negative 5 end cell 12 end table close parentheses open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 13 open parentheses table row 12 5 row cell negative 5 end cell 12 end table close parentheses to the power of negative 1 end exponent open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 13 open parentheses fraction numerator 1 over denominator 12 times 12 minus 5 times open parentheses negative 5 close parentheses end fraction open parentheses table row 12 cell negative 5 end cell row 5 12 end table close parentheses close parentheses open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 13 open parentheses fraction numerator 1 over denominator 144 plus 25 end fraction open parentheses table row 12 cell negative 5 end cell row 5 12 end table close parentheses close parentheses open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 13 open parentheses 1 over 169 open parentheses table row 12 cell negative 5 end cell row 5 12 end table close parentheses close parentheses open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 1 over 13 open parentheses table row 12 cell negative 5 end cell row 5 12 end table close parentheses open parentheses table row cell x to the power of apostrophe minus 3 end cell row cell y to the power of apostrophe minus 5 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 1 over 13 open parentheses table row cell 12 open parentheses x to the power of apostrophe minus 3 close parentheses minus 5 open parentheses y to the power of apostrophe minus 5 close parentheses end cell row cell 5 open parentheses x to the power of apostrophe minus 3 close parentheses plus 12 open parentheses y to the power of apostrophe minus 5 close parentheses end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 1 over 13 open parentheses table row cell 12 x to the power of apostrophe minus 36 minus 5 y to the power of apostrophe plus 25 end cell row cell 5 x to the power of apostrophe minus 15 plus 12 y to the power of apostrophe minus 60 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell row cell 1 over 13 open parentheses table row cell 12 x to the power of apostrophe minus 5 y to the power of apostrophe minus 11 end cell row cell 5 x to the power of apostrophe plus 12 y to the power of apostrophe minus 75 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 3 end cell row cell y minus 5 end cell end table close parentheses end cell end table end style

Sehingga didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 over 13 open parentheses 12 x to the power of apostrophe minus 5 y to the power of apostrophe minus 11 close parentheses end cell equals cell x minus 3 end cell row cell 12 over 13 x to the power of apostrophe minus 5 over 13 y to the power of apostrophe minus 11 over 13 end cell equals cell x minus 3 end cell row cell 12 over 13 x to the power of apostrophe minus 5 over 13 y to the power of apostrophe minus 11 over 13 plus 3 end cell equals x row cell 12 over 13 x to the power of apostrophe minus 5 over 13 y to the power of apostrophe plus 28 over 13 end cell equals x end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 over 13 open parentheses 5 x to the power of apostrophe plus 12 y to the power of apostrophe minus 75 close parentheses end cell equals cell y minus 5 end cell row cell 5 over 13 x to the power of apostrophe plus 12 over 13 y to the power of apostrophe minus 75 over 13 end cell equals cell y minus 5 end cell row cell 5 over 13 x to the power of apostrophe plus 12 over 13 y to the power of apostrophe minus 75 over 13 plus 5 end cell equals y row cell 5 over 13 x to the power of apostrophe plus 12 over 13 y to the power of apostrophe minus 10 over 13 end cell equals y end table end style

Titik A’(x’,y’) merupakan bayangan dari titik A(x,y). Karena titik A(x,y) terletak pada garis l: 12x + 5y = 36, maka titik A’(x’,y’) terletak pada bayangannya. Sehingga bentuklah garis tersebut ke dalam bentuk x’ dan y’, menggunakan hubungan yang telah didapat sebelumnya, yaitu begin mathsize 14px style x equals 12 over 13 x to the power of apostrophe minus 5 over 13 y to the power of apostrophe plus 28 over 13 end style dan begin mathsize 14px style y equals 5 over 13 x to the power of apostrophe plus 12 over 13 y to the power of apostrophe minus 10 over 13 end style.

Maka,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 12 x plus 5 y end cell equals 36 row cell 12 open parentheses 12 over 13 x to the power of apostrophe minus 5 over 13 y to the power of apostrophe plus 28 over 13 close parentheses plus 5 open parentheses 5 over 13 x to the power of apostrophe plus 12 over 13 y to the power of apostrophe minus 10 over 13 close parentheses end cell equals 36 row cell 144 over 13 x to the power of apostrophe minus 60 over 13 y to the power of apostrophe plus 336 over 13 plus 25 over 13 x to the power of apostrophe plus 60 over 13 y to the power of apostrophe minus 50 over 13 end cell equals 36 row cell 169 over 13 x to the power of apostrophe plus 286 over 13 end cell equals cell 3613 x to the power of apostrophe plus 22 equals 36 end cell row cell 13 x to the power of apostrophe end cell equals cell 36 minus 22 end cell row cell 13 x to the power of apostrophe end cell equals 14 row cell x to the power of apostrophe end cell equals cell 14 over 13 end cell end table end style

Sehingga bayangan dari garis l: 12x + 5y = 36 adalah garis begin mathsize 14px style k colon x equals 14 over 13 end style.
Jika garis k disubstitusi ke garis l, maka didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 12 open parentheses 14 over 13 close parentheses plus 5 y end cell equals 36 row cell 168 over 13 plus 5 y end cell equals 36 row cell 5 y end cell equals cell 36 minus 168 over 13 end cell row cell 5 y end cell equals cell 468 over 13 minus 168 over 13 end cell row cell 5 y end cell equals cell 300 over 13 end cell row y equals cell 60 over 13 end cell end table end style

Sehingga ordinat titik potong garis k dan l adalah begin mathsize 14px style 60 over 13 end style.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Tentukan bayangan: a.  jika dirotasi oleh

Pembahasan Soal:

Menentukan bayangan garis jika dirotasi oleh open square brackets open parentheses 1 comma 2 close parentheses comma space theta equals 30 degree close square brackets:

Kita akan melakukan dengan cara eliminasi Gauss-Jordan dalam menjawab soal ini.

Persamaan matriks Gauss-Jordan:

open parentheses table row cell cos space 30 degree end cell cell negative sin space 30 degree end cell row cell sin space 30 degree end cell cell cos space 30 degree end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses equals open parentheses table row cell 1 half square root of 3 end cell cell negative 1 half end cell row cell 1 half end cell cell 1 half square root of 3 end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses

Proses Eliminasi Gauss-Jordan:

left right double arrow open parentheses table row cell 1 half square root of 3 end cell cell negative 1 half end cell row cell 1 half end cell cell 1 half square root of 3 end cell end table vertical ellipsis table row cell x apostrophe minus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses table row cell fraction numerator 2 over denominator square root of 3 end fraction B subscript 2 end cell row cell fraction numerator 2 over denominator square root of 3 end fraction B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row cell fraction numerator square root of 3 over denominator 3 end fraction end cell 1 end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 2 square root of 3 over denominator 3 end fraction end cell row cell fraction numerator 2 square root of 3 over denominator 3 end fraction y apostrophe minus fraction numerator 4 square root of 3 over denominator 3 end fraction end cell end table close parentheses table row blank row cell negative fraction numerator square root of 3 over denominator 3 end fraction B subscript 1 plus B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row 0 cell 4 over 3 end cell end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 2 square root of 3 over denominator 3 end fraction end cell row cell negative 2 over 3 x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction y apostrophe plus 2 over 3 minus 4 over 3 square root of 3 end cell end table close parentheses table row blank row cell 3 over 4 B subscript 2 end cell end table left right double arrow open parentheses table row 1 cell negative fraction numerator square root of 3 over denominator 3 end fraction end cell row 0 1 end table vertical ellipsis table row cell fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe minus fraction numerator 8 square root of 3 over denominator 3 end fraction end cell row cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell end table close parentheses table row cell fraction numerator square root of 3 over denominator 3 end fraction B subscript 2 plus B subscript 1 end cell row blank end table left right double arrow open parentheses table row 1 0 row 0 1 end table vertical ellipsis table row cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction minus 1 end cell row cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell end table close parentheses

Hal ini berarti:

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 1 end cell equals cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction minus 1 end cell row x equals cell negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction space... left parenthesis 1 right parenthesis end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell y minus 2 end cell equals cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 1 half minus square root of 3 end cell row y equals cell negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3... left parenthesis 2 right parenthesis end cell end table

Kedua persamaan di atas disubstitusikan ke persamaan garis x square root of 3 minus y equals 0

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell x square root of 3 minus y end cell equals 0 row cell open parentheses negative fraction numerator square root of 3 over denominator 6 end fraction x apostrophe plus fraction numerator 2 square root of 3 over denominator 3 end fraction x apostrophe plus 1 half y apostrophe minus fraction numerator square root of 3 over denominator 6 end fraction minus fraction numerator 8 square root of 3 over denominator 3 end fraction close parentheses square root of 3 minus open parentheses negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3 close parentheses end cell equals 0 row cell open parentheses negative 1 half x apostrophe plus 2 x apostrophe plus fraction numerator square root of 3 over denominator 2 end fraction y apostrophe minus 1 half minus 8 close parentheses minus open parentheses negative 1 half x apostrophe plus 1 half square root of 3 y apostrophe plus 5 over 2 minus square root of 3 close parentheses end cell equals 0 row cell 2 x apostrophe minus 3 minus 8 minus square root of 3 end cell equals 0 row cell 2 x apostrophe minus 11 minus square root of 3 end cell equals 0 end table end style

Jadi, persamaan bayangan garis tersebut adalah 2 x apostrophe minus 11 minus square root of 3 equals 0.

 

 

0

Roboguru

Titik  diputar  berlawanan arah jarum jam dengan pusat putaran Titik  jika hasil rotasi adalah   maka nilai  adalah..

Pembahasan Soal:

Rotasi 90 degree berlawanan arah jarum jam mempunyai matriks straight M equals open parentheses table row 0 cell negative 1 end cell row 1 0 end table close parentheses dengan pusat perputaran open parentheses 1 comma 1 close parentheses, maka :

table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses table row cell straight x apostrophe end cell row cell straight y apostrophe end cell end table close parentheses end cell equals cell straight M open parentheses table row cell straight x minus 1 end cell row cell straight y minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell 2 plus straight a end cell row cell negative 2 end cell end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row 1 0 end table close parentheses open parentheses table row cell straight x minus 1 end cell row cell straight y minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell 2 plus straight a end cell row cell negative 2 end cell end table close parentheses end cell equals cell open parentheses table row cell straight a plus 1 end cell row cell 2 straight a minus 1 end cell end table close parentheses plus open parentheses table row 1 row 1 end table close parentheses end cell row cell open parentheses table row cell 2 plus straight a end cell row cell negative 2 end cell end table close parentheses end cell equals cell open parentheses table row cell straight a plus 2 end cell row cell 2 straight a end cell end table close parentheses end cell row cell negative 2 end cell equals cell 2 straight a space open parentheses kesamaan space matriks close parentheses end cell row straight a equals cell negative 1 end cell end table 

Jadi, nilai a adalah - 1

0

Roboguru

Jika titik  dirotasi oleh , koordinat bayangan titik  adalah .... (Kinomatika 2013)

Pembahasan Soal:

Diketahui,

  • straight A left parenthesis 2 comma space 4 right parenthesis
  • Dirotasikan oleh  begin mathsize 14px style open square brackets straight P open parentheses 4 comma space 6 close parentheses comma space straight R open parentheses 30 degree close parentheses close square brackets end style

Ditanyakan,

  • Koordinat bayangan titik straight A

Gunakan persamaan matriks berikut.

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos space straight theta end cell cell negative sin space straight theta end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close parentheses open parentheses table row cell x minus h end cell row cell y minus k end cell end table close parentheses plus open parentheses table row h row k end table close parentheses

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space 30 end cell cell negative sin space 30 end cell row cell sin space 30 end cell cell cos space 30 end cell end table close parentheses open parentheses table row cell 2 minus 4 end cell row cell 4 minus 6 end cell end table close parentheses plus open parentheses table row 4 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half square root of 3 end cell cell negative 1 half end cell row cell 1 half end cell cell 1 half square root of 3 end cell end table close parentheses open parentheses table row cell negative 2 end cell row cell negative 2 end cell end table close parentheses plus open parentheses table row 4 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell open parentheses 1 half square root of 3 times negative 2 close parentheses plus open parentheses negative 1 half times negative 2 close parentheses end cell row cell open parentheses 1 half times negative 2 close parentheses plus open parentheses 1 half square root of 3 times negative 2 close parentheses end cell end table close parentheses plus open parentheses table row 4 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell negative square root of 3 plus 1 end cell row cell negative 1 minus square root of 3 end cell end table close parentheses plus open parentheses table row 4 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell negative square root of 3 plus 5 end cell row cell 5 minus square root of 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 5 minus square root of 3 end cell row cell 5 minus square root of 3 end cell end table close parentheses end cell row blank blank blank end table

Jadi, koordinat  bayangan titik straight A adalah left parenthesis 5 minus square root of 3 comma space 5 minus square root of 3 right parenthesis.

Oleh karena itu, jawaban yang benar adalah E.

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