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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