# How do you use the Pythagorean theorem to find the missing side a given c=13, b=6a?

Feb 10, 2017

See the entire solution process below:

#### Explanation:

The Pythagorean Theorem states, for a right triangle:

${a}^{2} + {b}^{2} = {c}^{2}$

Substituting and solving using the values given in the problem for $b$ and $c$ results in:

${a}^{2} + {\left(6 a\right)}^{2} = {13}^{2}$

${a}^{2} + 36 {a}^{2} = 169$

$1 {a}^{2} + 36 {a}^{2} = 169$

$\left(1 + 36\right) {a}^{2} = 169$

$37 {a}^{2} = 169$

$\frac{37 {a}^{2}}{\textcolor{red}{37}} = \frac{169}{\textcolor{red}{37}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{37}}} {a}^{2}}{\cancel{\textcolor{red}{37}}} = 4.568$

${a}^{2} = 4.568$ rounded to the nearest thousandth.

$\sqrt{{a}^{2}} = \sqrt{4.568}$

$a = \sqrt{4.568} = 2.137$ rounded to the nearest thousandth.