Question 9
A paper equilateral triangle $ABC$ has side length $12.$ The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B.$ The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p.$
