# A triangle has sides A, B, and C. Sides A and B are of lengths 4 and 2, respectively, and the angle between A and B is (3pi)/4 . What is the length of side C?

Aug 1, 2016

$\left\mid C \right\mid = \textcolor{g r e e n}{2 \sqrt{5 + 2 \sqrt{2}}}$

#### Explanation:

If $c$ is the angle opposite side $C$
then $c = \frac{3 \pi}{4}$ (given)

and by the Law of Cosines
$\textcolor{w h i t e}{\text{XXX}} {C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos \left(c\right)$

In this case
$\textcolor{w h i t e}{\text{XXX}} {C}^{2} = {4}^{2} + {2}^{2} - 2 \cdot 4 \cdot 2 \cdot \left(- \frac{\sqrt{2}}{2}\right)$

$\textcolor{w h i t e}{\text{XXXX}} = 20 + 8 \sqrt{2}$

$\textcolor{w h i t e}{\text{XXXX}} = {2}^{2} \left(5 + 2 \sqrt{2}\right)$

$\Rightarrow$
$\textcolor{w h i t e}{\text{XXX}} \left\mid C \right\mid = 2 \sqrt{5 + 2 \sqrt{2}}$

(or using a calculator to evaluate an approximation)
$\textcolor{w h i t e}{\text{XXX}} \left\mid C \right\mid \approx 5.595865$