# a and b are the legs of a right triangle. c is the hypotenuse. If c=13, b=2a, what is a and b?

Feb 26, 2017

$a = \frac{13 \sqrt{5}}{5}$
$b = \frac{26 \sqrt{5}}{5}$

#### Explanation:

By Pythagorean Theorem, we know that:

${c}^{2} = {a}^{2} + {b}^{2}$, where $c$ is the hypotenuse and $a \mathmr{and} b$ are the legs.

Given that $c = 13 , \mathmr{and} b = 2 a$,

$\implies {13}^{2} = {a}^{2} + {\left(2 a\right)}^{2}$
$\implies {13}^{2} = {a}^{2} + 4 {a}^{2}$
$\implies {13}^{2} = 5 {a}^{2}$,
$\implies {a}^{2} = {13}^{2} / 5$
$\implies a = \frac{13}{\sqrt{5}} = \frac{13 \sqrt{5}}{5}$

$\implies b = 2 a = 2 \times \frac{13 \sqrt{5}}{5} = \frac{26 \sqrt{5}}{5}$