# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/8, the angle between sides B and C is (3pi)/4, and the length of side B is 7, what is the area of the triangle?

Feb 9, 2018

$17.324$

#### Explanation:

From the diagram:

$\boldsymbol{\theta} = \pi - \left(\frac{3 \pi}{4} + \frac{\pi}{8}\right) = \frac{\pi}{8}$

$\boldsymbol{\alpha} = \pi - \frac{3 \pi}{4} = \frac{\pi}{4}$

Using The Sine Rule

bb(SinA/a=SinB/b=SinC/c

We only need to find side $\boldsymbol{c}$

We know angle B and side b, so:

$\sin \frac{\frac{\pi}{8}}{7} = \sin \frac{\frac{\pi}{8}}{c} \implies c = \frac{7 \sin \left(\frac{\pi}{8}\right)}{\sin} \left(\frac{\pi}{8}\right) = 7$

From diagram:

$\boldsymbol{h} = 7 \sin \left(\frac{\pi}{4}\right)$

Area of triangle is:

$\boldsymbol{\frac{1}{2}}$base x height

$\frac{1}{2} c \times h$

$\frac{1}{2} \left(7\right) \cdot 7 \sin \left(\frac{\pi}{4}\right) = \frac{49 \sqrt{2}}{4} = 17.324$