Bouncing off of the Sides of A Triangle Forever

Geometry Level 5

Let \(ABC\) be a triangle such that \(BC=5\), \(CA=8\), and \(AB=9\). A ball is launched from a point \(X\) on segment \(BC\) such that it first bounces off of segment \(CA\) at a point \(Y\), then bounces off of segment \(AB\) at a point \(Z\), then bounces off of segment \(BC\) at \(X\), then bounces off of segment \(CA\) at \(Y\), and so on, traversing the perimeter of triangle \(XYZ\) forever.

The area of triangle \(XYZ\) can be written in the form \(\tfrac{p\sqrt{q}}{r}\), where \(p\) and \(r\) are coprime positive integers and \(q\) is a squarefree integer. Find \(p+q+r\).

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