# A triangle has sides A, B, and C. Sides A and B have lengths of 12 and 7, respectively. The angle between A and C is (7pi)/24 and the angle between B and C is  (5pi)/8. What is the area of the triangle?

Mar 26, 2018

color(indigo)("Area of the triangle "A_t ~~ 10.87 " sq units"

#### Explanation:

$a = 12 , b = 7 , \hat{A} = \frac{5 \pi}{8} , \hat{B} = \frac{7 \pi}{24}$

$\hat{C} = \pi - \hat{A} - \hat{B} = \pi - \frac{5 \pi}{8} - \frac{7 \pi}{24} = \frac{\pi}{12}$

Having known, two sides and the included angle (SAS), area of the triangle is given by the formula

color(indigo)(A_t = (1/2) a b sin C = (1/2) * 12 * 7 * sin (pi/12) ~~ 10.87 " sq units"