Answer
The third side would be a leg measuring $84$.
Work Step by Step
We can find the third side by using the Pythagorean theorem, which states that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the right triangle and $c$ is the hypotenuse.
Let's plug in what we know into the Pythagorean theorem:
$13^2 + 85^2 = c^2$
Evaluate the exponents:
$169 + 7225 = c^2$
Add to simplify:
$7394 = c^2$
If we take the square root of this number, the answer would not be rational, so we cannot be looking for the value of $c$. Let's see if $85$ may be the hypotenuse, so we may be looking for one of the legs of the right triangle.
Let's set up the equation such that $85$ becomes the hypotenuse:
$13^2 + b^2 = 85^2$
Evaluate the exponents:
$169 + b^2 = 7225$
Subtract $169$ from each side of the equation to move constants to one side of the equation:
$b^2 = 7056$
Take the positive square root to solve for $b$:
$b = 84$
The third side would be a leg measuring $84$.
