2006 AMC 10B Problems/Problem 19
Problem
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
?
Solution
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
.
Since ,
.
So, the area of triangle is
.
Therefore, the shaded area is