# A triangle has sides A, B, and C. Sides A and B have lengths of 8 and 1, respectively. The angle between A and C is (19pi)/24 and the angle between B and C is  (pi)/24. What is the area of the triangle?

Jul 29, 2016

$2$ sq.unit.

#### Explanation:

We know that the area of a $\Delta$ with sides A,B,C can be found by using any one of the following formulas :

Area$= \frac{1}{2} B C \sin \left(\hat{B , C}\right) = \frac{1}{2} C A \sin \left(\hat{C , A}\right) = \frac{1}{2} A B \sin \left(\hat{A , B}\right)$,

where, $\hat{A , B}$ denotes the angle btwn. sides $A \mathmr{and} B$, & likewise.

We are given the lengths of sides $A = 8 \mathmr{and} B = 1$, hence, the last formula will be more useful. Yes, that will require $\hat{A , B}$, which can be easily obtained by,

$\hat{A , B} = \pi - \hat{B , C} - \hat{C , A} = \pi - \left(\frac{\pi}{24} + 19 \frac{\pi}{24}\right) = 4 \frac{\pi}{24} = \frac{\pi}{6}$

Therefore, Area of the $\Delta = \frac{1}{2} \cdot 8 \cdot 1 \cdot \sin \left(\frac{\pi}{6}\right) = 4 \cdot \frac{1}{2} = 2$ sq.unit.