A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of 13 , 9 , and 3 , respectively. What is the rectangle's area?

Apr 25, 2018

$6 \sqrt{22}$

Explanation:

As is my practice, I'm converting to standard notation where triangle sides are small letters $a , b , c$. While we're naming things, let's call the other side of the rectangle $d$.

A right triangle has hypotenuse $a$ and sides $b$, $c$. Side $b$ forms a rectangle with another length $d$. Given $a = 13 , c = 9 , d = 3$ what is the area of the rectangle?

The area we seek is $b d$ so we have to find $b$, so this is all a long windup to a Pythagorean Theorem question:

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

$a$ is the hypotenuse so it's the one all by itself.

${b}^{2} = {a}^{2} - {c}^{2} = {13}^{2} - {9}^{2} = 169 - 81 = 88$

$b = \sqrt{88} = 2 \sqrt{22}$

$\textrm{a r e a} = b d = 6 \sqrt{22}$