Pertanyaan

Jika diketahui koordinat titik A(3, 1, 2), B(4, 3, 0), dan C(1, 2, 5), maka luas segitiga ABC sama dengan ....

R. Rohmat

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

Pembahasan

Cari panjang ruas garis AB, AC, dan BC terlebih dahulu. Sehingga didapat Kemudian Selanjutnya Sehingga perhatikan gambar segitiga berikut Melalui aturan cosinus, didapat bahwa Berdasarkan identitas trigonometri, maka didapat Karena 0° < B < 180°, maka sin B > 0 . Sehingga . Perhatikan bahwa luas segitiga ABC dapat dihitung sebagai berikut Jadi, jawaban yang tepat adalah B.

Cari panjang ruas garis AB, AC, dan BC terlebih dahulu. Sehingga didapat

Kemudian

Selanjutnya

Sehingga perhatikan gambar segitiga berikut

Melalui aturan cosinus, didapat bahwa

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell AC squared end cell equals cell AB squared plus B C squared minus sign 2 ⋅ AB ⋅ B C ⋅ cos invisible function application B end cell row cell open parentheses square root of 14 close parentheses squared end cell equals cell 3 squared plus open parentheses square root of 35 close parentheses squared minus sign 2 ⋅ 3 ⋅ square root of 35 ⋅ cos invisible function application B end cell row 14 equals cell 9 plus 35 minus sign 6 square root of 35 cos invisible function application B end cell row cell 6 square root of 35 cos invisible function application B end cell equals cell 9 plus 35 minus sign 14 end cell row cell 6 square root of 35 cos invisible function application B end cell equals 30 row cell cos invisible function application B end cell equals cell fraction numerator 30 over denominator 6 square root of 35 end fraction end cell row cell cos invisible function application B end cell equals cell fraction numerator 5 over denominator square root of 35 end fraction end cell row cell cos invisible function application B end cell equals cell fraction numerator 5 over denominator square root of 35 end fraction ⋅ fraction numerator square root of 35 over denominator square root of 35 end fraction end cell row cell cos invisible function application B end cell equals cell fraction numerator 5 square root of 35 over denominator 35 end fraction end cell row cell cos invisible function application B end cell equals cell fraction numerator square root of 35 over denominator 7 end fraction end cell end table end style$

Berdasarkan identitas trigonometri, maka didapat

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared invisible function application B plus cos squared invisible function application B end cell equals 1 row cell sin squared invisible function application B end cell equals cell 1 minus sign cos squared invisible function application B end cell row cell sin squared invisible function application B end cell equals cell 1 minus sign open parentheses fraction numerator square root of 35 over denominator 7 end fraction close parentheses squared end cell row cell sin squared invisible function application B end cell equals cell 1 minus sign 35 over 49 end cell row cell sin squared invisible function application B end cell equals cell 49 over 49 minus sign 35 over 49 end cell row cell sin squared invisible function application B end cell equals cell 14 over 49 end cell row cell sin invisible function application B end cell equals cell plus-or-minus fraction numerator square root of 14 over denominator 7 end fraction end cell end table end style$

Karena 0° < B < 180°, maka sin B > 0. Sehingga $begin mathsize 14px style sin invisible function application B equals fraction numerator square root of 14 over denominator 7 end fraction end style$ .

Perhatikan bahwa luas segitiga ABC dapat dihitung sebagai berikut

$begin mathsize 14px style L equals 1 half ⋅ AB ⋅ B C ⋅ sin invisible function application B L equals 1 half ⋅ 3 ⋅ square root of 35 ⋅ fraction numerator square root of 14 over denominator 7 end fraction L equals 3 over 14 square root of 490 L equals 3 over 14 ⋅ 7 square root of 10 L equals 3 over 2 square root of 10 end style$

Jadi, jawaban yang tepat adalah B.