Points \(P\) and \(Q\) are chosen on the two short sides of a right isosceles triangle, one on each side. The perpendiculars \(PM\) and \(QN\) are drawn such that \(M\) and \(N\) are points on the hypotenuse of the right isosceles triangle. Find the expected value of the area of trapezium \(PMNQ\), with proof.

**Bonus**: Generalise this for a right triangle with the two short sides of length \(a\) and \(b\).

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