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Pertanyaan

Pada segitiga ABC, angle A equals 75 degree comma angle B equals 60 degree,A C equals 8 square root of 3, dan sin invisible function application 105 degree equals 1 fourth open parentheses square root of 6 plus square root of 2 close parentheses. Luas segitiga ABC adalah ....

  1. 6 plus 2 square root of 3  

  2. 12 plus 4 square root of 3  

  3. 24 plus 8 square root of 3  

  4. 48 plus 16 square root of 3  

  5. 96 plus 32 square root of 3  

M. Robo

Master Teacher

Jawaban terverifikasi

Pembahasan

Berdasarkan informasi pada soal, didapat gambar segitiga sebagai berikut :

Ingat bahwa jumlah sudut dalam segitiga adalah undefined . Sehingga

angle A B C plus angle B C A plus angle C A B equals 180 degree 60 degree plus angle B C A plus 75 degree equals 180 degree angle B C A plus 135 degree equals 180 degree angle B C A equals box enclose 45 degree end enclose

Dengan aturan sinus, maka didapat

fraction numerator A B over denominator sin invisible function application angle B C A end fraction equals fraction numerator A C over denominator sin invisible function application angle A B C end fraction fraction numerator A B over denominator sin invisible function application 45 degree end fraction equals fraction numerator 8 square root of 3 over denominator sin invisible function application 60 degree end fraction fraction numerator A B over denominator 1 half square root of 2 end fraction equals fraction numerator 8 square root of 3 over denominator 1 half square root of 3 end fraction fraction numerator A B over denominator 1 half square root of 2 end fraction equals 16 A B equals 16 times 1 half square root of 2 A B equals box enclose 8 square root of 2 end enclose

Selanjutnya dengan menggunakan aturan sinus untuk luas segitiga, didapat bahwa

L equals 1 half times A B times A C times sin invisible function application angle C A B L equals 1 half times 8 square root of 2 times 8 square root of 3 times sin invisible function application 75 degree

Ingat bahwa

box enclose sin invisible function application open parentheses 180 degree minus alpha close parentheses equals sin invisible function application alpha end enclose

Sehingga jika alpha equals 75 degree didapat bahwa

sin invisible function application open parentheses 180 degree minus 75 degree close parentheses equals sin invisible function application 75 degree box enclose sin invisible function application 105 degree equals sin invisible function application 75 degree end enclose

Maka

sin invisible function application 75 degree equals sin invisible function application 105 degree equals box enclose 1 fourth open parentheses square root of 6 plus square root of 2 close parentheses end enclose

Sehingga

L equals 32 square root of 6 times sin invisible function application 75 degree L equals 32 square root of 6 times 1 fourth open parentheses square root of 6 plus square root of 2 close parentheses L equals 8 square root of 6 open parentheses square root of 6 plus square root of 2 close parentheses L equals 8 square root of 6 times square root of 6 plus 8 square root of 6 times square root of 2 L equals 8 times 6 plus 8 square root of 12 L equals 48 plus 8 times 2 square root of 3 L equals box enclose 48 plus 16 square root of 3 end enclose    

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