Kalkulator Segitiga

Selesaikan segitiga, jika diketahui $$$a = 9$$$, $$$b = 9 \sqrt{2}$$$, $$$C = 45^{\circ}$$$.

Menurut hukum kosinus: $$$c^{2} = a^{2} + b^{2} - 2 a b \cos{\left(C \right)}$$$.

Dalam kasus ini, $$$c^{2} = 9^{2} + \left(9 \sqrt{2}\right)^{2} - \left(2\right)\cdot \left(9\right)\cdot \left(9 \sqrt{2}\right)\cdot \left(\cos{\left(45^{\circ} \right)}\right) = 81.$$$

Dengan demikian, $$$c = 9$$$.

Menurut hukum sinus: $$$\frac{a}{\sin{\left(A \right)}} = \frac{c}{\sin{\left(C \right)}}$$$.

Dalam kasus ini, $$$\frac{9}{\sin{\left(A \right)}} = \frac{9}{\sin{\left(45^{\circ} \right)}}$$$.

Dengan demikian, $$$\sin{\left(A \right)} = \frac{\sqrt{2}}{2}$$$.

Terdapat dua kemungkinan kasus:

1. $$$A = 45^{\circ}$$$

   Sudut ketiga adalah $$$B = 180^{\circ} - \left(A + C\right)$$$.

   Dalam kasus ini, $$$B = 180^{\circ} - \left(45^{\circ} + 45^{\circ}\right) = 90^{\circ}$$$.

   Luasnya adalah $$$S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(9\right)\cdot \left(9 \sqrt{2}\right)\cdot \left(\sin{\left(45^{\circ} \right)}\right) = \frac{81}{2}.$$$

   Kelilingnya adalah $$$P = a + b + c = 9 + 9 \sqrt{2} + 9 = 9 \left(\sqrt{2} + 2\right)$$$.

2. $$$A = 135^{\circ}$$$

   Sudut ketiga adalah $$$B = 180^{\circ} - \left(A + C\right)$$$.

   Dalam kasus ini, $$$B = 180^{\circ} - \left(135^{\circ} + 45^{\circ}\right) = 0^{\circ}$$$.

   Kasus ini tidak mungkin, karena sudutnya tidak positif.

$$$a = 9$$$A

$$$b = 9 \sqrt{2}\approx 12.727922061357855$$$A

$$$c = 9$$$A

$$$A = 45^{\circ}$$$A

$$$B = 90^{\circ}$$$A

$$$C = 45^{\circ}$$$A

Luas: $$$S = \frac{81}{2} = 40.5$$$A.

Keliling: $$$P = 9 \left(\sqrt{2} + 2\right)\approx 30.727922061357855$$$A.