2012 UNCO Math Contest II Problems/Problem 7
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.
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
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