# A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 10, respectively. The angle between A and C is (pi)/8 and the angle between B and C is  (3pi)/8. What is the area of the triangle?

Jan 7, 2017

$S = 35$

#### Explanation:

If $A \angle C = \frac{\pi}{8}$ and $B \angle C = \frac{3 \pi}{8}$ then as the sum of the internal angles of a triangle always equals $\pi$, we have:

$A \angle B = \pi - \frac{\pi}{8} - \frac{3 \pi}{8} = \pi - \frac{4 \pi}{8} = \pi - \frac{\pi}{2} = \frac{\pi}{2}$

Thus this is a right triangle where $A$ and $B$ are the legs, and $C$ is the hypotenuse.

So the area of the triangle is:

$S = \frac{A \cdot B}{2} = \frac{7 \cdot 10}{2} = \frac{70}{2} = 35$