Question 11
Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $\overline{AB},$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of $\triangle CDM$ can be expressed as $\frac{m\sqrt{n}}{p},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m + n + p.$