How do you use the pythagorean theorem to find the missing side given c = 19, b = 4a; find a and b?

Jan 18, 2017

$a = 4.608$ rounded to the nearest thousandth.

and

$b = 18.433$ rounded to the nearest thousandth.

Explanation:

The Pythagorean theorem states:

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

Where $a$ and $b$ are the lengths of the sides of a right triangle and $c$ is the length of the hypotenuse of the right triangle.

Susbtituting $\textcolor{red}{19}$ for $c$ and $\textcolor{b l u e}{4 a}$ for $b$ we can solve for $a$:

${a}^{2} + {\left(\textcolor{b l u e}{4 a}\right)}^{2} = {\textcolor{red}{19}}^{2}$

a^2 + 16a^2 = 361

$17 {a}^{2} = 361$

$\frac{17 {a}^{2}}{\textcolor{red}{17}} = \frac{361}{\textcolor{red}{17}}$

(color(red)(cancel(color(black)(17)))a^2)/cancel(color(red)(17)) = 21.235

${a}^{2} = 21.235$ rounded to the nearest thousandth.

$\sqrt{{a}^{2}} = \sqrt{21.235}$

$a = 4.608$ rounded to the nearest thousandth.

Then we can solve for $b$ by substituting $4.608$ for $a$ in the relationship: $b = 4 a$

$b = 4 \times 4.608$

$b = 18.433$ rounded to the nearest thousandth.