Bonne Chance!

Calculus Level 3

A triangle and a rectangle are in semicircles of radius rr where two of their vertices are on the diameter, and the rests are on the arc. The angle of vertex AA of the triangle is equally probable to be any angle between 00 and π2\dfrac { \pi }{ 2 }, while for a rectangle the height GHGH is equally probable to be any height between 00 and rr. 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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