A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 8, 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?

Jan 3, 2016

$\text{Area} = 4 \sqrt{3}$ sq. units exactly or 6.9 sq. units (1 decimal place )

Explanation:

I recommend that you draw a sketch of the triangle indicating the lengths of sides and angles given in the question.

Require the angle between A and B which is the third angle in the triangle and since the sum of all 3 angles = $\pi$

then

$\text{3rd angle} = \pi - 11 \frac{\pi}{24} - 5 \frac{\pi}{24}$

$= 24 \frac{\pi}{24} - 16 \frac{\pi}{24}$

$= 8 \frac{\pi}{24}$

$= \frac{\pi}{3}$

using

$\text{Area} = \frac{1}{2} \times A \times B \times \sin \left(\frac{\pi}{3}\right)$

$\text{Area} = \frac{1}{2} \times 2 \times 8 \times \frac{\sqrt{3}}{2}$

$= 8 \times \frac{\sqrt{3}}{2}$

$= 4 \sqrt{3} {\text{ units"^2 ~~ "6.9 units}}^{2}$ ( 1 decimal place )