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Pentagon and Acute Probability (Posted on 2016-08-02) Difficulty: 3 of 5
A point O is selected at random from the interior of the pentagon with vertices
P = (0, 2), Q = (4, 0), R = (2pi + 1, 0), S = (2pi + 1, 4), and T = (0, 4).

Determine the probability that ∠POQ is acute.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 1 of 1
The area of pentagon PQRST is that of rectangle Origin,RST minus that of triangle Origin,QP.

Angle POQ is acute if O lies outside the semicircle with diameter PQ. That semicircle falls entirely within the given pentagon, so we need only subtract the area of that semicircle from the area of the pentagon to get the area in which O must lie for a success (acute angle).

Pentagon PQRST has area 4*(2*pi+1) - 4 = 8*pi.

The radius of the semicircle of obtuseness is sqrt(5) The area of a full circle would be 5*pi, so the semicircle has area 5*pi/2.

The requested probability is (8*pi - 5*pi/2) / (8*pi) = 1 - 5/(2*8) = 1 - 5/16 = 11/16.

  Posted by Charlie on 2016-08-02 11:54:27
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