# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/12, the angle between sides B and C is (5pi)/6, and the length of side B is 9, what is the area of the triangle?

##### 1 Answer
Dec 15, 2015

$\frac{81}{4}$

#### Explanation:

We use ABC for points; and a,b,c for opposite sides.

angle between a and b = $\hat{C} = \frac{1}{12} \pi$

angle between b and c = $\hat{A} = \frac{5}{6} \pi$

$\hat{B} = \pi - \hat{C} - \hat{A} = \pi \left(1 - \frac{1}{12} - \frac{5}{6}\right) = \frac{1}{12} \pi = \hat{C}$

Therefore, $c = b = 9$.

Let $H = \frac{B + C}{2.}$

We want ${S}_{\Delta} = \frac{1}{2} \cdot a \cdot h$

$\sin \hat{C} = \frac{h}{9}$ AND $\cos \hat{C} = \frac{a}{18}$

${S}_{\Delta} = \frac{1}{2} \cdot 18 \cos \hat{C} \cdot 9 \sin \hat{C} = \frac{81}{2} \sin \frac{\pi}{6} = \frac{81}{2} \cdot \frac{1}{2}$