Question

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?

Answer 1

The area of the triangle is `30 " units"^2`.

Explanation:

Let the angle between `A` and `C` be `beta`, the angle between `B` and `C` be `alpha` and finally, the angle between `A` and `B` be `gamma`.

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 `180^@ = pi`.

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 `A` and `B`. This makes the calculation of the area easy:

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