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Calculus Level 3

A triangle and a rectangle are in semicircles of radius \(r\) where two of their vertices are on the diameter, and the rests are on the arc. The angle of vertex \(A\) of the triangle is equally probable to be any angle between \(0\) and \(\dfrac { \pi }{ 2 }\), while for a rectangle the height \(GH\) is equally probable to be any height between \(0\) and \(r\). If you throw a dart and hit a polygon in the semicircle you will win a prize, but you are blindfolded, and you just have one chance. Which polygon will you choose? Why?

Assume that the dart has an equal chance of landing anywhere on the semicircle.

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