# An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of 8  and the triangle has an area of 36 . What are the lengths of sides A and B?

May 23, 2018

$\left\mid A \right\mid = \left\mid B \right\mid \approx 6.02$ (units)

#### Explanation:

If ${\triangle}_{A B C}$ has a base of side $C$ and $\left\mid A \right\mid = \left\mid B \right\mid$
then, given an area of $36$ (square units),
the triangle must have a height (relative to the base $C$)
$\textcolor{w h i t e}{\text{XXX}} h = \frac{36}{\left\mid C \right\mid} = \frac{36}{8} = 4.5$

Dividing the triangle into two right angled triangles as in the image below:

and using the Pythagorean Theorem:
$\textcolor{w h i t e}{\text{XXX}} \left\mid A \right\mid = \left\mid B \right\mid = \sqrt{{4}^{2} + {\left(4.5\right)}^{2}} = \frac{\sqrt{145}}{2} \approx 6.02$