2012 UNCO Math Contest II Problems/Problem 7
Problem
A circle of radius is externally tangent to a circle of radius
and both circles are tangent to a line. Find the area of the shaded
region that lies between the two circles and the line.
Solution
If we draw in the second tangent between these circles, we get a point where the tangents intersect. Drawing a line through the centers of the circle and the point where the tangents meet, we get two similar triangles in ratio . Let the hypotenuse of the smaller triangle be
. Since the distance between the centers of the circles is
, we can write the ratio
. Solving for
, we get
. Since the hypotenuse of the smaller triangle is
, and one of the legs is
, we see that it is a
triangle. So the side lengths of the smaller triangle is
and the side lengths of the larger triangle is
. Finding the difference between the areas of both triangles, we get
which is the area of the trapezoid. The trapezoid is the area of a
sector of the smaller circle, a
sector of the larger circle, and the shaded region. Subtracting the areas of both sectors from the area of the trapezoid, we get
~Ultraman
See Also
2012 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |