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Let P be a point inside square ABCD.  Find the probability that angle APB is acute.

 Sep 7, 2020
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I love this problem! Construct Semicircle AB. Note that if the triangle is in Semicircle AB, it is obtuse. Note that if P is on the circumference of AB, the triangle is right. Note that if the triangle is outside AB, it is acute. Assume that ABCD is a unit square (WLOG). The area of the square is 1. The area of the semicircle is \(\frac{\pi}{8}\). This means that the answer is \(1-\frac{\pi}{8}=\boxed{\frac{8-\pi}{8}}\)

 Sep 7, 2020

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