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Garis x + y = 5 dirotasi sebesar 45° berlawanan arah jarum jam dengan pusat P(1,-3). Bayangan dari garis tersebut adalah ....

Pertanyaan

Garis x + y = 5 dirotasi sebesar 45° berlawanan arah jarum jam dengan pusat P(1,-3). Bayangan dari garis tersebut adalah ....

  1. begin mathsize 14px style y equals fraction numerator 6 square root of 2 minus 7 over denominator 2 end fraction end style 

  2. begin mathsize 14px style y equals fraction numerator 7 square root of 2 minus 6 over denominator 2 end fraction end style 

  3. begin mathsize 14px style x equals fraction numerator 6 square root of 2 minus 7 over denominator 2 end fraction end style 

  4. begin mathsize 14px style x equals fraction numerator 7 square root of 2 minus 6 over denominator 2 end fraction end style 

  5. begin mathsize 14px style x equals fraction numerator 6 square root of 2 plus 7 over denominator 2 end fraction end style 

Pembahasan:

Misalkan titik A(x, y) terletak pada garis tersebut.
Kemudian titik tersebut juga dirotasikan terhadap pusat (1,-3). Sehingga a = 1 dan b = -3.
Besar sudut rotasinya adalah 45° berlawanan arah jarum jam. Sehingga θ = 45°.
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 45 degree end cell cell negative sin invisible function application 45 degree end cell row cell sin invisible function application 45 degree end cell cell cos invisible function application 45 degree end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus open parentheses negative 3 close parentheses end cell end table close parentheses plus open parentheses table row 1 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 cell 1 half square root of 2 end cell cell negative 1 half square root of 2 end cell row cell 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 cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses plus open parentheses table row 1 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 minus open parentheses table row 1 row cell negative 3 end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half square root of 2 end cell cell negative 1 half square root of 2 end cell row cell 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 cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe plus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half square root of 2 end cell cell negative 1 half square root of 2 end cell row cell 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 cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell 1 half square root of 2 end cell cell negative 1 half square root of 2 end cell row cell 1 half square root of 2 end cell cell 1 half square root of 2 end cell end table close parentheses to the power of negative 1 end exponent open parentheses table row cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe plus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell fraction numerator 1 over denominator 1 half square root of 2 times 1 half square root of 2 minus open parentheses negative 1 half square root of 2 close parentheses times 1 half square root of 2 end fraction 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 cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe plus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell fraction numerator 1 over denominator 1 half plus 1 half end fraction 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 cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe plus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell 1 over 1 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 cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe plus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell 1 half square root of 2 open parentheses x to the power of apostrophe minus 1 close parentheses plus 1 half square root of 2 open parentheses y to the power of apostrophe plus 3 close parentheses end cell row cell negative 1 half square root of 2 open parentheses x to the power of apostrophe minus 1 close parentheses plus 1 half square root of 2 open parentheses y to the power of apostrophe plus 3 close parentheses end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell 1 half x to the power of apostrophe square root of 2 minus 1 half square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 3 over 2 square root of 2 end cell row cell negative 1 half x to the power of apostrophe square root of 2 plus 1 half square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 3 over 2 square root of 2 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus square root of 2 end cell row cell negative 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 2 square root of 2 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y plus 3 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 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus square root of 2 end cell equals cell x minus 1 end cell row x equals cell 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus square root of 2 plus 1 end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 2 square root of 2 end cell equals cell y plus 3 end cell row y equals cell negative 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 2 square root of 2 minus 3 end cell end table end style

Titik A’(x’,y’) merupakan bayangan dari titik A(x,y). Karena titik A(x,y) terletak pada garis x + y = 5, maka titik A’(x’,y’) terletak pada bayangannya. Sehingga bentuklah parabola tersebut ke dalam bentuk x’ dan y’, menggunakan hubungan yang telah didapat sebelumnya, yaitu begin mathsize 14px style x equals 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus square root of 2 plus 1 space d a n space y equals negative 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 2 square root of 2 minus 3 end style.

Maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x plus y end cell equals 5 row cell open parentheses 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus square root of 2 plus 1 close parentheses plus open parentheses negative 1 half x to the power of apostrophe square root of 2 plus 1 half y to the power of apostrophe square root of 2 plus 2 square root of 2 minus 3 close parentheses end cell equals 5 row cell y to the power of apostrophe square root of 2 plus 3 square root of 2 minus 2 end cell equals 5 row cell y to the power of apostrophe square root of 2 end cell equals cell 5 minus 3 square root of 2 plus 2 end cell row cell y to the power of apostrophe square root of 2 end cell equals cell 7 minus 3 square root of 2 end cell row cell y to the power of apostrophe end cell equals cell fraction numerator 7 minus 3 square root of 2 over denominator square root of 2 end fraction end cell row cell y to the power of apostrophe end cell equals cell fraction numerator 7 minus 3 square root of 2 over denominator square root of 2 end fraction times fraction numerator square root of 2 over denominator square root of 2 end fraction end cell row cell y to the power of apostrophe end cell equals cell fraction numerator 7 square root of 2 minus 6 over denominator 2 end fraction end cell end table end style

Dalam bentuk umum didapat garis bayangannya adalah begin mathsize 14px style y equals fraction numerator 7 square root of 2 minus 6 over denominator 2 end fraction end style.

Jawaban terverifikasi

Dijawab oleh:

R. Munawaroh

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 10 Januari 2021

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