More Wrecked Angles

Geometry Level 4

In the diagram above, ABCD and PQRS are both rectangles. Points P, Q, R, and S lie on segments \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\), respectively, and \(\overline{BQ} < \overline{QC}\).

If \(AB=36\) and \(BC=50\), then the maximum possible value of \(BQ\) can be written in the form \(a-\sqrt{b}\), where \(a,b \in \mathbb{N}\). What is \(a+b\)?

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