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Salinlah gambar berikut pada kertas berpetak. Gambarkan bangun hasil △ABC sesuai data di bawah ini. d. [P,R(−180∘)]

Pertanyaan

Salinlah gambar berikut pada kertas berpetak.

Gambarkan bangun hasil begin mathsize 14px style triangle ABC end style sesuai data di bawah ini.

d. begin mathsize 14px style open square brackets straight P comma space straight R open parentheses negative 180 degree close parentheses close square brackets end style 

Pembahasan Soal:

Gambar bangun hasil begin mathsize 14px style triangle ABC end style dengan  begin mathsize 14px style open square brackets straight P comma space straight R open parentheses negative 180 degree close parentheses close square brackets end style 


Jadi,gambar bangun hasil begin mathsize 14px style triangle ABC end style dengan begin mathsize 14px style open square brackets straight P comma space straight R open parentheses negative 180 degree close parentheses close square brackets end style seperti di atas.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Jika garis l dirotasikan terhadap pusat (−2, 1) sejauh 90° berlawanan arah jarum jam, maka bayangan garisnya adalah x + 3y + 4 = 0. Persamaan dari garis l adalah ....

Pembahasan Soal:

Misalkan titik A(x, y) terletak pada garis l.
Kemudian titik tersebut juga dirotasikan terhadap pusat (-2,1). Sehingga a = undefined2 dan b = 1.
Besar sudut rotasinya adalah 90° berlawanan arah jarum jam. Sehingga θ = 90°.
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 theta end cell cell negative sin invisible function application theta end cell row cell sin invisible function application theta end cell cell cos invisible function application theta 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 cos invisible function application 90 degree end cell cell negative sin invisible function application 90 degree end cell row cell sin invisible function application 90 degree end cell cell cos invisible function application 90 degree end cell end table close parentheses open parentheses table row cell x minus open parentheses negative 2 close parentheses end cell row cell y minus 1 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 1 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 0 cell negative 1 end cell row 1 0 end table close parentheses open parentheses table row cell x plus 2 end cell row cell y minus 1 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 1 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 0 times open parentheses x plus 2 close parentheses plus open parentheses negative 1 close parentheses times open parentheses y minus 1 close parentheses end cell row cell 1 times open parentheses x plus 2 close parentheses plus 0 times open parentheses y minus 1 close parentheses end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 1 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 negative y plus 1 end cell row cell x plus 2 end cell end table close parentheses plus open parentheses table row cell negative 2 end cell row 1 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 open parentheses negative y plus 1 close parentheses plus open parentheses negative 2 close parentheses end cell row cell open parentheses x plus 2 close parentheses plus 1 end cell 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 negative y minus 1 end cell row cell x plus 3 end cell end table close parentheses end cell end table end style

Sehingga didapatkan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x to the power of apostrophe end cell equals cell negative y minus 1 end cell row cell y to the power of apostrophe end cell equals cell x plus 3 end cell end table end style

Titik (x’,y’) ini merupakan titik yang terletak pada garis bayangannya, yaitu x + 3y + 4 = 0. Sehingga dapat dituliskan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x to the power of apostrophe plus 3 y to the power of apostrophe plus 4 end cell equals 0 row cell open parentheses negative y minus 1 close parentheses plus 3 open parentheses x plus 3 close parentheses plus 4 end cell equals 0 row cell negative y minus 1 plus 3 x plus 9 plus 4 end cell equals 0 row cell 3 x minus y plus 12 end cell equals 0 end table end style

Jika kedua ruas dikalikan dengan undefined1, maka didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 x plus y minus 12 end cell equals 0 row cell y minus 3 x minus 12 end cell equals 0 end table end style

Maka didapatkan persamaan dari garis l yaitu yundefined3xundefined12 = 0.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Diketahui △PQR dengan P(−3,3), Q(4,6), dan R(2,−3). Tentukan bayangan (peta)  yang dirotasikan terhadap titik berat  sejauh 2π​ searah jarum jam.

Pembahasan Soal:

  • Menentukan titik berat segitiga P Q R:

Z left parenthesis a comma b right parenthesis equals open parentheses fraction numerator P subscript x plus Q subscript x plus R subscript x over denominator 2 end fraction comma fraction numerator P subscript y plus Q subscript y plus R subscript y over denominator 2 end fraction close parentheses Z left parenthesis a comma b right parenthesis equals open parentheses fraction numerator left parenthesis negative 3 right parenthesis plus 4 plus 2 over denominator 2 end fraction comma fraction numerator 3 plus 6 plus left parenthesis negative 3 right parenthesis over denominator 2 end fraction close parentheses Z left parenthesis a comma b right parenthesis equals open parentheses 3 over 2 comma 3 close parentheses

Jadi, triangle P Q R dirotasi terhadap Z open parentheses 3 over 2 comma 3 close parentheses sejauh straight pi over 2 searah jarum jam.

  • Menentukan bayangan triangle P Q R:

Titik P':

open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space 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 open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row cell cos space open parentheses negative straight pi over 2 close parentheses end cell cell negative sin space open parentheses negative straight pi over 2 close parentheses end cell row cell sin space open parentheses negative straight pi over 2 close parentheses end cell cell cos space open parentheses negative straight pi over 2 close parentheses end cell end table close parentheses open parentheses table row cell negative 3 minus 3 over 2 end cell row cell 3 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell negative 3 minus 3 over 2 end cell row cell 3 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell negative 9 over 2 end cell row 0 end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row 0 row cell 9 over 2 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript p apostrophe end cell row cell y subscript p apostrophe end cell end table close parentheses equals open parentheses table row cell 3 over 2 end cell row cell 15 over 2 end cell end table close parentheses

Titik Q apostrophe:

open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space 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 open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row cell cos space open parentheses negative straight pi over 2 close parentheses end cell cell negative sin space open parentheses negative straight pi over 2 close parentheses end cell row cell sin space open parentheses negative straight pi over 2 close parentheses end cell cell cos space open parentheses negative straight pi over 2 close parentheses end cell end table close parentheses open parentheses table row cell 4 minus 3 over 2 end cell row cell 6 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell 4 minus 3 over 2 end cell row cell 6 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell 5 over 2 end cell row 3 end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row 3 row cell negative 5 over 2 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript Q apostrophe end cell row cell y subscript Q apostrophe end cell end table close parentheses equals open parentheses table row cell 9 over 2 end cell row cell 1 half end cell end table close parentheses

Titik R apostrophe:

open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space 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 open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row cell cos space open parentheses negative straight pi over 2 close parentheses end cell cell negative sin space open parentheses negative straight pi over 2 close parentheses end cell row cell sin space open parentheses negative straight pi over 2 close parentheses end cell cell cos space open parentheses negative straight pi over 2 close parentheses end cell end table close parentheses open parentheses table row cell 2 minus 3 over 2 end cell row cell negative 3 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell 2 minus 3 over 2 end cell row cell negative 3 minus 3 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row cell 1 half end cell row cell negative 6 end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row cell negative 6 end cell row cell negative 1 half end cell end table close parentheses plus open parentheses table row cell 3 over 2 end cell row 3 end table close parentheses open parentheses table row cell x subscript R apostrophe end cell row cell y subscript R apostrophe end cell end table close parentheses equals open parentheses table row cell negative 9 over 2 end cell row cell 5 over 2 end cell end table close parentheses

Jadi, bayangan triangle P Q R space adalah P apostrophe open parentheses 3 over 2 comma 15 over 2 close parentheses comma space Q apostrophe open parentheses 9 over 2 comma 1 half close parentheses comma space R apostrophe open parentheses negative 9 over 2 comma space 5 over 2 close parentheses.

0

Roboguru

Titik H(3,−2) dirotasikan sebesar 150° terhadap titik (−1,3). Hasilnya dirotasikan lagi sebesar 120° terhadap titik pusat (−1,−3). Hasil akhir rotasi tit!R H adalah . . .

Pembahasan Soal:

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 open parentheses 150 degree plus 120 degree close parentheses end cell cell negative sin open parentheses 150 degree plus 120 degree close parentheses end cell row cell sin open parentheses 150 degree plus 120 degree close parentheses end cell cell cos open parentheses 150 degree plus 120 degree close parentheses end cell end table close parentheses open parentheses table row cell 3 minus open parentheses negative 1 close parentheses end cell row cell negative 2 minus open parentheses negative 3 close parentheses end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 3 end cell 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 0 1 row cell negative 1 end cell 0 end table close parentheses open parentheses table row 4 row 1 end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 3 end cell end table close parentheses equals open parentheses table row 0 row cell negative 7 end cell end table close parentheses end cell end table end style

Jadi, hasil akhir rotasi adalah E. begin mathsize 14px style bold H bold " bold left parenthesis bold 0 bold comma bold minus bold 7 bold right parenthesis end style

0

Roboguru

Tentukan bayangan: b. kurva≡x2−4y=0 jika dirotasi oleh [(−1,0),θ=90∘]

Pembahasan Soal:

  • Menentukan bayanga kurva x squared minus 4 y equals 0 dirotasi oleh open square brackets open parentheses negative 1 comma space 0 close parentheses comma space theta equals 90 degree close square brackets

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 90 degree end cell cell negative sin space 90 degree end cell row cell sin space 90 degree end cell cell cos space 90 degree 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 open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row 0 cell negative 1 end cell row 1 0 end table close parentheses open parentheses table row cell x plus 1 end cell row cell y minus 0 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row 0 end table close parentheses 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 negative y end cell row cell x plus 1 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row 0 end table close parentheses 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 negative y minus 1 end cell row cell x plus 1 end cell end table close parentheses

diperoleh:

x apostrophe equals negative y minus 1 rightwards arrow y equals negative x apostrophe minus 1 space... left parenthesis 1 right parenthesis space space space y apostrophe equals x plus 1 rightwards arrow x equals y apostrophe minus 1 space space space space space space space space space... left parenthesis 2 right parenthesis

Kedua persamaan tersebut disubstitusikan ke persamaan kurva:

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 4 y end cell equals 0 row cell open parentheses y apostrophe minus 1 close parentheses squared minus 4 open parentheses negative x apostrophe minus 1 close parentheses end cell equals 0 row cell open parentheses y apostrophe minus 1 close parentheses squared plus 4 left parenthesis x apostrophe plus 1 right parenthesis end cell equals 0 row cell open parentheses y apostrophe minus 1 close parentheses squared end cell equals cell negative 4 left parenthesis x apostrophe plus 1 right parenthesis end cell end table

Jadi, bayangan kurva tersebut adalah open parentheses y minus 1 close parentheses squared equals negative 4 open parentheses x apostrophe plus 1 close parentheses.

0

Roboguru

Lingkaran dengan persamaan x2+y2−6x=4 dirotasikan terhadap pusat P(5,0) sebesar  searah putaran jarum jam.

Pembahasan Soal:

Dengan menerapkan konsep rotasi dengan titik pusat begin mathsize 14px style straight P left parenthesis straight a comma space straight b right parenthesis end style sebesar begin mathsize 14px style theta end style searah putaran jarum jam, yaitu:

begin mathsize 14px style open parentheses table row cell straight x apostrophe end cell row cell straight 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 times open parentheses table row cell straight x minus straight a end cell row cell straight y minus straight b end cell end table close parentheses plus open parentheses table row straight a row straight b end table close parentheses end style 

Dengan:

begin mathsize 14px style straight a equals 5 comma space straight b equals 0 space dan space straight theta equals 270 degree end style.

Misalkan begin mathsize 14px style left parenthesis straight x comma space straight y right parenthesis end style adalah titik-titik yang dilalui oleh lingkaran begin mathsize 14px style straight x squared plus straight y squared minus 6 straight x equals 4 end style maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px 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 open parentheses table row cell cos space 270 degree end cell cell negative sin space 270 degree end cell row cell sin space 270 degree end cell cell cos space 270 degree end cell end table close parentheses times open parentheses table row cell straight x minus 5 end cell row cell straight y minus 0 end cell end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row 0 cell negative left parenthesis negative 1 right parenthesis end cell row cell negative 1 end cell 0 end table close parentheses times open parentheses table row cell straight x minus 5 end cell row straight y end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row 0 1 row cell negative 1 end cell 0 end table close parentheses times open parentheses table row cell straight x minus 5 end cell row straight y end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row cell 0 times left parenthesis straight x minus 5 right parenthesis plus 1 times left parenthesis straight y right parenthesis end cell row cell negative 1 left parenthesis straight x minus 5 right parenthesis plus 0 times straight y end cell end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row straight y row cell negative 1 left parenthesis straight x minus 5 right parenthesis end cell end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row straight y row cell negative straight x plus 5 end cell end table close parentheses plus open parentheses table row 5 row 0 end table close parentheses end cell row blank equals cell open parentheses table row cell straight y plus 5 end cell row cell negative straight x plus 5 end cell end table close parentheses end cell end table end style 

Dengan konsep kesamaan matriks diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px 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 open parentheses table row cell straight y plus 5 end cell row cell negative straight x plus 5 end cell end table close parentheses end cell row cell straight x apostrophe end cell equals cell straight y plus 5 end cell row cell straight x apostrophe minus 5 end cell equals straight y row straight y equals cell straight x apostrophe minus 5 end cell row cell straight y apostrophe end cell equals cell negative straight x plus 5 end cell row straight x equals cell negative straight y apostrophe plus 5 end cell end table end style  

maka bayangan begin mathsize 14px style straight x squared plus straight y squared minus 6 straight x equals 4 end style oleh rotasi tersebut adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis negative straight y apostrophe plus 5 right parenthesis squared plus left parenthesis straight x apostrophe minus 5 right parenthesis squared minus 6 left parenthesis negative straight y apostrophe plus 5 right parenthesis end cell equals 4 row cell straight y apostrophe squared minus 10 straight y apostrophe plus 25 plus straight x apostrophe squared minus 10 straight x apostrophe plus 25 plus 6 straight y apostrophe minus 30 end cell equals 4 row cell straight x apostrophe squared plus straight y apostrophe squared minus 10 straight x apostrophe minus 4 straight y apostrophe plus 20 end cell equals 4 row cell straight x apostrophe squared plus straight y apostrophe squared minus 10 straight x apostrophe minus 4 straight y apostrophe plus 16 end cell equals 0 row blank blank blank end table end style 

Jadi, Bayangan lingkaran begin mathsize 14px style straight x squared plus straight y squared minus 6 straight x equals 4 end style jika dirotasikan terhadap pusat Error converting from MathML to accessible text. sebesar size 14px 270 size 14px degree searah putaran jarum jam adalah begin mathsize 14px style straight x squared plus straight y squared minus 10 straight x minus 4 straight y plus 16 equals 0 end style.

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Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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