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
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 |