Question 7
Triangle $ABC$ has side lengths $AB = 12,$ $BC = 25,$ and $CA = 17.$ Rectangle $PQRS$ has vertex $P$ on $\overline{AB},$ vertex $Q$ on $\overline{AC},$ and vertices $R$ and $S$ on $\overline{BC}.$ In terms of the side length $PQ = w,$ the area of $PQRS$ can be expressed as the quadratic polynomial $$
\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.
$$ Then the coefficient $\beta = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\text{Area}(PQRS) = \alpha w - \beta \cdot w^2.
$$ Then the coefficient $\beta = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$