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Pertanyaan

Diketahui segitiga ABC dengan titik sudut masing-masing A ( 2 , 1 ) , B ( 4 , 2 ) dan C ( 2 , 4 ) . Tentukan: c. Gambar segitiga dan bayangan segitiga pada rotasi 4 5 ∘ searah jarum jam dengan pusat rotasi P ( − 1 , − 1 ) .

Diketahui segitiga  dengan titik sudut masing-masing ,  dan . Tentukan:

c. Gambar segitiga ABC dan bayangan segitiga ABC pada rotasi  searah jarum jam dengan pusat rotasi .

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Y. Fathoni

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Jawaban terverifikasi

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Pembahasan

Rotasi terhadap titik asal , dengan sudut rotasi , dapat dihitung dengan rumus berikut. Diketahui segitiga dengan titik sudut masing-masing , dan . Akan ditentukan gambar segitiga dan bayangan segitiga pada rotasi searah jarum jam dengan pusat rotasi . Karena rotasi searah jarum jam, maka Rotasi bernilai negatif. Menentukan bayangan titik . Menentukan bayangan titik . Menentukan bayangan titik . Jadi, diperoleh titik koordinat bayangan segitiga adalah , dan . Jika digambarkan akan menjadi seperti berikut.

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 parentheses, straight 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 45 degree searah jarum jam dengan pusat rotasi straight P open parentheses negative 1 comma space minus 1 close parentheses. Karena rotasi searah jarum jam, maka Rotasi bernilai negatif.

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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 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 negative 45 degree close parentheses end cell cell negative sin space open parentheses negative 45 degree close parentheses end cell row cell sin space open parentheses negative 45 degree close parentheses end cell cell cos space open parentheses negative 45 degree close parentheses end cell end table close parentheses open parentheses table row cell 2 minus open parentheses negative 1 close parentheses end cell row cell 1 minus open parentheses negative 1 close parentheses end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell cos space 45 degree end cell cell sin space 45 degree end cell row cell negative sin space 45 degree end cell cell cos space 45 degree end cell end table close parentheses open parentheses table row cell 2 plus 1 end cell row cell 1 plus 1 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half square root of 2 end cell cell 1 half square root of 2 end cell row cell negative 1 half square root of 2 end cell cell 1 half square root of 2 end cell end table close parentheses open parentheses table row 3 row 2 end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 3 over 2 square root of 2 plus 2 over 2 square root of 2 end cell row cell negative 3 over 2 square root of 2 plus 2 over 2 square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 5 over 2 square root of 2 end cell row cell negative 1 half square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 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 5 over 2 square root of 2 minus 1 end cell row cell negative 1 half square root of 2 minus 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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 degree close parentheses close square brackets on top straight A apostrophe open parentheses 5 over 2 square root of 2 minus 1 comma space minus 1 half square root of 2 minus 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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 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 negative 45 degree close parentheses end cell cell negative sin space open parentheses negative 45 degree close parentheses end cell row cell sin space open parentheses negative 45 degree close parentheses end cell cell cos space open parentheses negative 45 degree close parentheses end cell end table close parentheses open parentheses table row cell 4 minus open parentheses negative 1 close parentheses end cell row cell 2 minus open parentheses negative 1 close parentheses end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell cos space 45 degree end cell cell sin space 45 degree end cell row cell negative sin space 45 degree end cell cell cos space 45 degree end cell end table close parentheses open parentheses table row cell 4 plus 1 end cell row cell 2 plus 1 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half square root of 2 end cell cell 1 half square root of 2 end cell row cell negative 1 half square root of 2 end cell cell 1 half square root of 2 end cell end table close parentheses open parentheses table row 5 row 3 end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 5 over 2 square root of 2 plus 3 over 2 square root of 2 end cell row cell negative 5 over 2 square root of 2 plus 3 over 2 square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 over 2 square root of 2 end cell row cell negative 2 over 2 square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 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 4 square root of 2 minus 1 end cell row cell negative square root of 2 minus 1 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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 degree close parentheses close square brackets on top straight B apostrophe open parentheses 4 square root of 2 minus 1 comma space minus square root of 2 minus 1 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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 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 negative 45 degree close parentheses end cell cell negative sin space open parentheses negative 45 degree close parentheses end cell row cell sin space open parentheses negative 45 degree close parentheses end cell cell cos space open parentheses negative 45 degree close parentheses end cell end table close parentheses open parentheses table row cell 2 minus open parentheses negative 1 close parentheses end cell row cell 4 minus open parentheses negative 1 close parentheses end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell cos space 45 degree end cell cell sin space 45 degree end cell row cell negative sin space 45 degree end cell cell cos space 45 degree end cell end table close parentheses open parentheses table row cell 2 plus 1 end cell row cell 4 plus 1 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 half square root of 2 end cell cell 1 half square root of 2 end cell row cell negative 1 half square root of 2 end cell cell 1 half square root of 2 end cell end table close parentheses open parentheses table row 3 row 5 end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 3 over 2 square root of 2 plus 5 over 2 square root of 2 end cell row cell negative 3 over 2 square root of 2 plus 5 over 2 square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 8 over 2 square root of 2 end cell row cell 2 over 2 square root of 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row cell negative 1 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 4 square root of 2 minus 1 end cell row cell square root of 2 minus 1 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 negative 1 comma space minus 1 close parentheses comma space straight R open parentheses negative 45 degree close parentheses close square brackets on top straight C apostrophe open parentheses 4 square root of 2 minus 1 comma space square root of 2 minus 1 close parentheses end style

Jadi, diperoleh titik koordinat bayangan segitiga ABC adalah straight A apostrophe open parentheses 5 over 2 square root of 2 minus 1 comma space minus 1 half square root of 2 minus 1 close parentheses, straight B apostrophe open parentheses 4 square root of 2 minus 1 comma space minus square root of 2 minus 1 close parentheses dan straight C apostrophe open parentheses 4 square root of 2 minus 1 comma space square root of 2 minus 1 close parentheses.

Jika digambarkan akan menjadi seperti berikut.

Latihan Bab

Transformasi Translasi (Pergeseran)

Transformasi Refleksi

Transformasi Rotasi

Transformasi Dilatasi

Perdalam pemahamanmu bersama Master Teacher
di sesi Live Teaching, GRATIS!

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