A rectangle has coordinates A (0, 0), B (0, 2), C ( ,2), D( , 0)
A point, P, is chosen at random within the rectangle.
What is the probability that the angle APB is obtuse?
Imagine that P lies on a semicircle whose diameter is AB.
Then angle APB will be exactly 90°.
For any P within that semicircle angle APB will be obtuse and for any P outside that semicircle, angle APB will be acute.
Hence the probability that angle APB is obtuse is given by the area of the semicircle divided by the area of the rectangle.
Area rectangle =
Probability that APB is obtuse = (Area semicircle) (Area rectangle) =