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.
![[asy] draw((-1,-1)--(7,-1),black); filldraw((0,-1)--(0,0)--(2*sqrt(3),2)--(2*sqrt(3),-1)--cycle,grey); filldraw(circle((0,0),1),white); filldraw(circle((2*sqrt(3),2),3),white); [/asy]](http://latex.artofproblemsolving.com/2/1/7/217135aee72ffa973880c2216bf966bda2b48c44.png)
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 | ||
