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Area of Rectangle Inscribed in Triangle

Source: 2015 AIME 2 Problem 7

March 26, 2015
geometryrectangle

Problem Statement

Triangle ABCABC has side lengths AB=12AB=12, BC=25BC=25, and CA=17CA=17. Rectangle PQRSPQRS has vertex PP on AB‾\overline{AB}, vertex QQ on AC‾\overline{AC}, and vertices RR and SS on BC‾\overline{BC}. In terms of the side length PQ=wPQ=w, the area of PQRSPQRS can be expressed as the quadratic polynomial Area(PQRS)=αw−β⋅w2\text{Area}(PQRS)=\alpha w-\beta\cdot w^2 Then the coefficient β=mn\beta=\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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