# A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 2, 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?

Dec 9, 2015

Area $\triangle A B C = 7$

#### Explanation:

If $\angle \left(C : A\right) = \frac{5 \pi}{24}$
and
$\angle \left(B : C\right) = \frac{7 \pi}{24}$

then
$\textcolor{w h i t e}{\text{XXX}} \angle \left(A : B\right) = \pi - \left(\frac{5 \pi}{24} + \frac{7 \pi}{24}\right)$

$\textcolor{w h i t e}{\text{XXXXXXXX}} = \pi - \frac{12 \pi}{24} = \frac{\pi}{2}$

i.e. $\angle \left(A : B\right)$ is a right angle

Treating $A$ as the base and $B$ as the height:

$\textcolor{w h i t e}{\text{XXX}} A r e a = \frac{1}{2} \times b a s e \times h e i g h t = \frac{1}{2} \times 7 \times 2 = 7$