# A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 6, respectively. The angle between A and C is (pi)/8 and the angle between B and C is  (pi)/24. What is the area of the triangle?

Mar 7, 2016

This triangle is not possible.

#### Explanation:

We begin by attempting to draw the triangle in question. Let's start with the vertex for the first angle that we were given. This is the angle between sides A and C which is $\pi / 8$ where we know that the length of A is 3 units.

We know that side B has length 6 units - twice as long as side A. It becomes clear that to draw B, it must be at a large angle to A, which makes sense from the angle we got for the vertex between B and C which is a very small angle $\pi / 24$. In fact we know that the angles in a triangle add up to $\pi$ so the unknown angle must be:

$\theta = \pi - \frac{\pi}{8} - \frac{\pi}{24} = \frac{5 \pi}{6}$

Attempting to complete the triangle using a side of 6 units at the prescribed angles:

We come to realize that this is not a possible triangle - there is no way to make side B meet side C. Either one of the angles given is wrong or one of the lengths is wrong.