2006 AMC 10B Problems/Problem 19
A circle of radius is centered at . Square has side length . Sides and are extended past to meet the circle at and , respectively. What is the area of the shaded region in the figure, which is bounded by , , and the minor arc connecting and ?
The shaded area is equivilant to the area of sector minus the area of triangle plus the area of triangle .
Using the Pythagorean Theorem:
Clearly and are triangles with
Since is a square,
can be found by doing some subtraction of angles.
So the area of sector is
The area of triangle is
So the area of triangle is
Therefore, the shaded area is