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

##### 1 Answer

Jan 26, 2016

The area of the triangle is

#### Explanation:

Let the angle between

We already know that

#beta = (5 pi)/24 = 37.5^@#

and

#alpha = (7 pi)/24 = 52.5^@#

We also know that the sum of the angles of the triangle must be

Thus, we can compute the third angle:

#gamma = pi - (5pi)/24 - (7pi)/24 = pi - (12 pi)/24 = pi/2 = 90^@#

This means that the triangle has a right angle between

#"area" = 1/2 A * B * color(grey) (underbrace(sin(pi/2))_(=1)) = 1/2 * 12 * 5 = 30 #