# 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 56  and the triangle has an area of 112 . What are the lengths of sides A and B?

May 12, 2018

$20 \sqrt{2}$

#### Explanation:

Let's renotate and say triangle ABC has $a = b$, $c = 56$ area $A = 112$.

Archimedes' Theorem is a modern version of Heron's Formula,

$16 {A}^{2} = 4 {a}^{2} {c}^{2} - \left({b}^{2} - {a}^{2} - {c}^{2}\right)$

$16 {\left(112\right)}^{2} = 4 {a}^{2} {\left(56\right)}^{2} - {56}^{4}$

$a = 20 \sqrt{2}$