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PQRS pada gambar di atas adalah layang-layang. Diagonalnya berpotongan di titik O. Jika panjang PO=36​ cm, PR=86​ cm, dan QS=82​ cm, hitunglah: b. Keliling layang-layang .

Pertanyaan


text PQRS end text pada gambar di atas adalah layang-layang. Diagonalnya berpotongan di titik text O end text. Jika panjang text PO end text equals 3 square root of 6 space text cm end text comma space text PR end text equals 8 square root of 6 space text cm end text comma space text dan QS end text equals 8 square root of 2 space text cm end text, hitunglah:
b. Keliling layang-layang text PQRS end text.

R. Febrianti

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

keliling layang-layang tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 open parentheses square root of 84 plus square root of 182 close parentheses space text cm end text end cell end table.

Pembahasan

Diketahui layang-layang text PQRS end text dengan panjang text PR end text equals 8 square root of 6 space text cm end text comma space text dan QS end text equals 8 square root of 2 space text cm end text seperti pada gambar di bawah ini.

Diketahui text PO end text equals 3 square root of 6 space text cm dan end text space text PR end text equals 8 square root of 6 space text cm end text sehingga panjang text OR end text yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell text OR end text end cell equals cell text PR end text minus text PO end text end cell row blank equals cell 8 square root of 6 minus 3 square root of 6 end cell row blank equals cell open parentheses 8 minus 3 close parentheses square root of 6 end cell row blank equals cell 5 square root of 6 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang yaitu salah satu diagonalnya membagi diagonal yang lain sama besar maka text QO end text equals text OS end text equals fraction numerator 8 square root of 2 over denominator 2 end fraction equals 4 square root of 2 space text cm end text.

Berdasarkan karakteristik layang-layang yaitu kedua diagonalnya saling berpotongan tegak lurus maka angle text POQ end text merupakan sudut siku-siku sehingga terbentuk segitiga siku-siku seperti pada gambar di bawah ini.

Sehingga panjang text PQ end text dapat ditentukan dengan menggunakan teorema pythagoras sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat dengan aturan sebagai berikut.

n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space text dengan end text space n not equal to 0

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 3 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell end table

Berdasarkan sifat bilangan berpangkat yaitu open parentheses a cross times b close parentheses to the power of m equals a to the power of m cross times b to the power of m dan open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponentmaka

table attributes columnalign right center left columnspacing 0px end attributes row cell text PQ end text end cell equals cell square root of text PO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 3 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 3 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 3 squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared plus 4 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 9 cross times 6 to the power of 1 half cross times 2 end exponent plus 16 cross times 2 to the power of 1 half cross times 2 end exponent end root end cell row blank equals cell square root of 9 cross times 6 to the power of 1 plus 16 cross times 2 to the power of 1 end root end cell row blank equals cell square root of 9 cross times 6 plus 16 cross times 2 end root end cell row blank equals cell square root of 54 plus 32 end root end cell row blank equals cell square root of 86 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang maka text PS end text equals text PQ end text equals square root of 86 space text cm end text

Berdasarkan karakteristik layang-layang yaitu kedua diagonalnya saling berpotongan tegak lurus maka angle text ROQ end text merupakan sudut siku-siku sehingga terbentuk segitiga siku-siku seperti pada gambar di bawah ini.

Sehingga panjang text RQ end text dapat ditentukan dengan menggunakan teorema pythagoras sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell end table

Bentuk akar dapat diubah menjadi bilangan berpangkat dengan aturan sebagai berikut.

n-th root of a to the power of m end root equals a to the power of m over n end exponent comma space text dengan end text space n not equal to 0

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 5 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell end table

Berdasarkan sifat bilangan berpangkat yaitu open parentheses a cross times b close parentheses to the power of m equals a to the power of m cross times b to the power of m dan open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponentmaka

table attributes columnalign right center left columnspacing 0px end attributes row cell text RQ end text end cell equals cell square root of text RO end text squared plus text OQ end text squared end root end cell row blank equals cell square root of open parentheses 5 square root of 6 close parentheses squared plus open parentheses 4 square root of 2 close parentheses squared end root end cell row blank equals cell square root of open parentheses 5 cross times 6 to the power of 1 half end exponent close parentheses squared plus open parentheses 4 cross times 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 5 squared cross times open parentheses 6 to the power of 1 half end exponent close parentheses squared plus 4 squared cross times open parentheses 2 to the power of 1 half end exponent close parentheses squared end root end cell row blank equals cell square root of 25 cross times 6 to the power of 1 half cross times 2 end exponent plus 16 cross times 2 to the power of 1 half cross times 2 end exponent end root end cell row blank equals cell square root of 25 cross times 6 to the power of 1 plus 16 cross times 2 to the power of 1 end root end cell row blank equals cell square root of 9 cross times 6 plus 16 cross times 2 end root end cell row blank equals cell square root of 150 plus 32 end root end cell row blank equals cell square root of 182 space text cm end text end cell end table

Berdasarkan karakteristik layang-layang maka text RS end text equals text RQ end text equals square root of 182 space text cm end text. Keliling layang-layang dapat ditentukan dengan menjumlahkan semua sisi-sisinya sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell text K end text end cell equals cell text PQ end text plus text PS end text plus text RQ end text plus text RS end text end cell row blank equals cell square root of 84 plus square root of 84 plus square root of 182 plus square root of 182 end cell row blank equals cell 2 square root of 84 plus 2 square root of 182 end cell row blank equals cell 2 open parentheses square root of 84 plus square root of 182 close parentheses space text cm end text end cell end table

Jadi keliling layang-layang tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 open parentheses square root of 84 plus square root of 182 close parentheses space text cm end text end cell end table.

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