# A triangle has sides A, B, and C. Sides A and B have lengths of 12 and 5, respectively. The angle between A and C is (5pi)/24 and the angle between B and C is  (7pi)/24. What is the area of the triangle?

Jan 26, 2016

The area of the triangle is $30 {\text{ units}}^{2}$.

#### Explanation:

Let the angle between $A$ and $C$ be $\beta$, the angle between $B$ and $C$ be $\alpha$ and finally, the angle between $A$ and $B$ be $\gamma$.

$\beta = \frac{5 \pi}{24} = {37.5}^{\circ}$

and

$\alpha = \frac{7 \pi}{24} = {52.5}^{\circ}$

We also know that the sum of the angles of the triangle must be ${180}^{\circ} = \pi$.

Thus, we can compute the third angle:

$\gamma = \pi - \frac{5 \pi}{24} - \frac{7 \pi}{24} = \pi - \frac{12 \pi}{24} = \frac{\pi}{2} = {90}^{\circ}$

This means that the triangle has a right angle between $A$ and $B$. This makes the calculation of the area easy:

$\text{area} = \frac{1}{2} A \cdot B \cdot \textcolor{g r e y}{{\underbrace{\sin \left(\frac{\pi}{2}\right)}}_{= 1}} = \frac{1}{2} \cdot 12 \cdot 5 = 30$