# A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 4, respectively. The angle between A and C is (11pi)/24 and the angle between B and C is  (5pi)/24. What is the area of the triangle?

Jul 19, 2016

Area$= 7 \sqrt{3} s q . u n i t . \cong 12.1244 s q . u n i t .$#

#### Explanation:

Let us denote, by $\hat{A , B}$, the angle btwn. sides A and B.

$\hat{A , B} = \pi - \left\{\hat{A , C} + \hat{C , B}\right\} = \pi - \left(11 \frac{\pi}{24} + 5 \frac{\pi}{24}\right) = \pi - 16 \frac{\pi}{24} = 8 \frac{\pi}{24}$

$\therefore \angle \left(A , B\right) = \frac{\pi}{3}$

Now, from Trigo., we know that,

Area of $\Delta$

$= \frac{1}{2} A \cdot B \cdot \sin \left(\hat{A , B}\right) = \frac{1}{2} \cdot 7 \cdot 4 \cdot \sin \left(\frac{\pi}{3}\right) = 14 \cdot \frac{\sqrt{3}}{2} = 7 \sqrt{3}$

Taking, $\sqrt{3} \cong 1.7321 , \Delta \cong 12.1244 s q . u n i t .$