Fix a line segment L in the plane.
Among all triangles which have L as longest side one is chosen at random.
What is the probability it is obtuse?
pi/2 - 1
Let L=1. x and y and the other legs. Then x<=1, y<=1, which is the unit square in the first quadrant. The triangle is obtuse if and only if x^2 + y^2 < 1, which means (x, y) is within the circle in the first quadrant.
But to form a triangle, x+y>1. Therefore, the probability is
(pi/4 - 1/2)/(1/2) = pi/2 - 1
Edited on October 25, 2017, 1:46 pm
Posted by chun
on 2017-10-25 13:24:12