# AIME 2015 Problem 7

Geometry Level 4

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

Area($$PQRS$$) = $$a w - b w^2$$.

Then the coefficient $$b = \dfrac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m+n$$.

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