Question

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?

Answer 1

`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 `bd` 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}`

`text{area} = bd = 6 sqrt{22}`