# A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is (11pi)/24 and the angle between B and C is  (7pi)/24. What is the area of the triangle?

Feb 21, 2016

${\text{Area}}_{\triangle} = \approx 12.97$

#### Explanation:

Call the angle opposite side A as $\angle a$ (i.e. the angle between B and C);
and similarly the angle opposite Side B as $\angle b$
and the angle opposite Side C as $\angle c$

We are told (among other things) that:
$A = 7 , \angle a = \frac{7 \pi}{24} , \mathmr{and} \angle c = \frac{11 \pi}{24}$

The Sine Law tells us
$\textcolor{w h i t e}{\text{XXX}} \frac{C}{\sin} \left(c\right) = \frac{A}{\sin} \left(a\right)$

So
$\textcolor{w h i t e}{\text{XXX}} C = \frac{7}{\sin \left(\frac{7 \pi}{24}\right)} \cdot \sin \left(\frac{11 \pi}{24}\right)$

Evaluating we get: $C = 11$

We now have the lengths of the three sides and can apply Heron's Formula:
color(white)("XXX")"Area"_triangle = sqrt(S(A-A)(S-B)(S-C))
where $S$ is the semi-perimeter (i.e. $\frac{A + B + C}{2}$) 