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Diketahui suatu garis l:4x−3y=12 dirotasikan sejauh α berlawanan arah jarum jam dengan pusat (1,4) sehingga diperoleh garis k. Jika tanα=−34​ dengan interval 90° < α < 180°. Maka titik potong antara garis l dan garis k adalah ....

Pertanyaan

Diketahui suatu garis begin mathsize 14px style l space colon space 4 x minus 3 y equals 12 end style dirotasikan sejauh α berlawanan arah jarum jam dengan pusat (1,4) sehingga diperoleh garis k. Jika begin mathsize 14px style tan invisible function application alpha equals negative 4 over 3 end style dengan interval 90° < α < 180°. Maka titik potong antara garis l dan garis k adalah ....

  1. (8, 9)

  2. (9, 8)

  3. (undefined8, 9)

  4. (undefined9, 8)

  5. (8, undefined9)

Pembahasan:

Perhatikan bahwa begin mathsize 14px style tan invisible function application alpha equals negative 4 over 3 end style dengan interval 90° < α < 180°.
Karena 90° < α < 180°, maka α berada di kuadran II.
Misalkan β = 180° undefinedα, 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 tan invisible function application beta end cell equals cell tan invisible function application open parentheses 180 degree minus alpha close parentheses end cell row cell tan invisible function application beta end cell equals cell negative tan invisible function application alpha end cell row cell tan invisible function application beta end cell equals cell negative open parentheses negative 4 over 3 close parentheses end cell row cell tan invisible function application beta end cell equals cell 4 over 3 end cell end table end style

Karena begin mathsize 14px style tan invisible function application beta equals 4 over 3 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 4 over x equals 4 over 5 space d a n space cos invisible function application beta equals 3 over x equals 3 over 5 end style.
Perhatikan bahwa

begin mathsize 14px style sin invisible function application beta equals sin invisible function application open parentheses 180 degree minus alpha close parentheses equals sin invisible function application alpha end style

Sehingga

begin mathsize 14px style sin invisible function application alpha equals sin invisible function application beta equals 4 over 5 end style

dan

begin mathsize 14px style cos invisible function application beta equals cos invisible function application open parentheses 180 degree minus alpha close parentheses equals negative cos invisible function application alpha end style

Sehingga

begin mathsize 14px style cos invisible function application alpha equals negative cos invisible function application beta equals negative 3 over 5 end style

Selanjutnya misalkan titik A(x,y) terletak pada garis l.
Kemudian titik tersebut juga dirotasikan terhadap pusat (1, 4). Sehingga a = 1 dan b = 4.
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 negative 3 over 5 end cell cell negative 4 over 5 end cell row cell 4 over 5 end cell cell negative 3 over 5 end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses plus open parentheses table row 1 row 4 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 fifth open parentheses table row cell negative 3 end cell cell negative 4 end cell row 4 cell negative 3 end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses plus open parentheses table row 1 row 4 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 4 end table close parentheses end cell equals cell 1 fifth open parentheses table row cell negative 3 end cell cell negative 4 end cell row 4 cell negative 3 end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus 4 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 minus 4 end cell end table close parentheses equals 1 fifth open parentheses table row cell negative 3 end cell cell negative 4 end cell row 4 cell negative 3 end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 5 open parentheses table row cell x to the power of apostrophe minus 1 end cell row cell y to the power of apostrophe minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell negative 3 end cell cell negative 4 end cell row 4 cell negative 3 end cell end table close parentheses open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 5 open parentheses table row cell negative 3 end cell cell negative 4 end cell row 4 cell negative 3 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 minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 5 open parentheses fraction numerator 1 over denominator open parentheses negative 3 close parentheses times open parentheses negative 3 close parentheses minus open parentheses negative 4 close parentheses times 4 end fraction open parentheses table row cell negative 3 end cell 4 row cell negative 4 end cell cell negative 3 end cell end table close parentheses 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 minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 5 open parentheses fraction numerator 1 over denominator 9 plus 16 end fraction open parentheses table row cell negative 3 end cell 4 row cell negative 4 end cell cell negative 3 end cell end table close parentheses 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 minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 5 open parentheses 1 over 25 open parentheses table row cell negative 3 end cell 4 row cell negative 4 end cell cell negative 3 end cell end table close parentheses 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 minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 1 fifth open parentheses table row cell negative 3 end cell 4 row cell negative 4 end cell cell negative 3 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 minus 4 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 1 fifth open parentheses table row cell negative 3 open parentheses x to the power of apostrophe minus 1 close parentheses plus 4 open parentheses y to the power of apostrophe minus 4 close parentheses end cell row cell negative 4 open parentheses x to the power of apostrophe minus 1 close parentheses minus 3 open parentheses y to the power of apostrophe minus 4 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 minus 4 end cell end table close parentheses end cell row cell 1 fifth open parentheses table row cell negative 3 x to the power of apostrophe plus 3 plus 4 y to the power of apostrophe minus 16 end cell row cell negative 4 x to the power of apostrophe plus 4 minus 3 y to the power of apostrophe plus 12 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell row cell 1 fifth open parentheses table row cell negative 3 x to the power of apostrophe plus 4 y to the power of apostrophe minus 13 end cell row cell negative 4 x to the power of apostrophe minus 3 y to the power of apostrophe plus 16 end cell end table close parentheses end cell equals cell open parentheses table row cell x minus 1 end cell row cell y minus 4 end cell end table close parentheses end cell end table end style

Sehingga dapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 fifth open parentheses negative 3 x to the power of apostrophe plus 4 y to the power of apostrophe minus 13 close parentheses end cell equals cell x minus 1 end cell row cell negative 3 over 5 x to the power of apostrophe plus 4 over 5 y to the power of apostrophe minus 13 over 5 end cell equals cell x minus 1 end cell row cell negative 3 over 5 x to the power of apostrophe plus 4 over 5 y to the power of apostrophe minus 13 over 5 plus 1 end cell equals x row cell negative 3 over 5 x to the power of apostrophe plus 4 over 5 y to the power of apostrophe minus 8 over 5 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 fifth open parentheses negative 4 x to the power of apostrophe minus 3 y to the power of apostrophe plus 16 close parentheses end cell equals cell y minus 4 end cell row cell negative 4 over 5 x to the power of apostrophe minus 3 over 5 y to the power of apostrophe plus 16 over 5 end cell equals cell y minus 4 end cell row cell negative 4 over 5 x to the power of apostrophe minus 3 over 5 y to the power of apostrophe plus 16 over 5 plus 4 end cell equals y row cell negative 4 over 5 x to the power of apostrophe minus 3 over 5 y to the power of apostrophe plus 36 over 5 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: 4x - 3y = 12, 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 negative 3 over 5 x to the power of apostrophe plus 4 over 5 y to the power of apostrophe minus 8 over 5 space d a n space y equals negative 4 over 5 x to the power of apostrophe minus 3 over 5 y to the power of apostrophe plus 36 over 5 end style.
Maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x minus 3 y end cell equals 12 row cell 4 open parentheses negative 3 over 5 x to the power of apostrophe plus 4 over 5 y to the power of apostrophe minus 8 over 5 close parentheses minus 3 open parentheses negative 4 over 5 x to the power of apostrophe minus 3 over 5 y to the power of apostrophe plus 36 over 5 close parentheses end cell equals 12 row cell negative 12 over 5 x to the power of apostrophe plus 16 over 5 y to the power of apostrophe minus 32 over 5 plus 12 over 5 x to the power of apostrophe plus 9 over 5 y to the power of apostrophe minus 108 over 5 end cell equals 12 row cell 25 over 5 y to the power of apostrophe minus 140 over 5 end cell equals 12 row cell 5 y to the power of apostrophe minus 28 end cell equals 12 row cell 5 y to the power of apostrophe end cell equals cell 12 plus 28 end cell row cell 5 y to the power of apostrophe end cell equals 40 row cell y to the power of apostrophe end cell equals 8 end table end style

Sehingga bayangan dari garis l: 4x - 3y = 12 adalah garis k: y = 8.
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 4 x minus 3 y end cell equals 12 row cell 4 x minus 3 open parentheses 8 close parentheses end cell equals 12 row cell 4 x minus 24 end cell equals 12 row cell 4 x end cell equals cell 12 plus 24 end cell row cell 4 x end cell equals 36 row x equals 9 end table end style

Sehingga koordinat titik potong garis k dan l adalah (9,8).

Jadi, jawaban yang tepat adalah B.

Jawaban terverifikasi

Dijawab oleh:

I. Sutiawan

Mahasiswa/Alumni Universitas Pasundan

Terakhir diupdate 04 Oktober 2021

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