Find the area of the shaded region in the above diagram.
The area of the shaded region is equal to the area of the semicircle with diameter plus the area of the semicircle with diameter plus the area of the right triangle minus the area of the semicircle with diameter . We have
is the diameter of a circle with center at . If and , find the measure of .
Draw line . Then by Thales' theorem, is a right triangle.
Since is isosceles with ,
By the inscribed angle theorem,
is the diameter of the circle with center at is perpendicular to . If and , find
Since , we have
Substituting (1) into (2) gives
Using the Pythagorean theorem, we have
Rationalizing the denominator gives