Answer
$m=\sqrt{p^2-n^2}$
Work Step by Step
Using $a^2+b^2=c^2$ or the Pythagorean Theorem, the relationship of the sides of the given right triangle is
\begin{align*}
m^2+n^2&=p^2
.\end{align*}
Using the properties of equality to isolate $m^2$ results to
\begin{align*}
m^2&=p^2-n^2
.\end{align*}
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{align*}
m&=\pm\sqrt{p^2-n^2}
.\end{align*}
Since $m$ should be greater than zero and $p$ is the longest side (i.e. $p$ is the hypotenuse), then
\begin{align*}
m&=\sqrt{p^2-n^2}
.\end{align*}
