Problem:
A point P is selected at random from the interior of the pentagon with vertices A=(0,2),B=(4,0),C=(2π+1,0),D=(2π+1,4), and E=(0,4). What is the probability that ∠APB is obtuse?
Answer Choices:
A. 51​
B. 41​
C. 165​
D. 83​
E. 21​ Solution:
Since ∠APB=90∘ if and only if P lies on the semicircle with center (2,1) and radius 5​, the angel is obtuse if and only if the point P lies inside this semicircle. The semicircle lies entirely inside the pentagon, since the distance, 3 , from (2,1) to DE is greater than the radius of the circle. Thus the probability that the nagle is obtuse
is the ratio of the area of the semicircle to the area of the pentagon.
Let O=(0,0),A=(0,2),B=(4,0),C=(2Ï€+1,0),D=(2Ï€+1,4), and E=(0,4). Then the area of the pentagon is
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