Mock Geometry AIME 2011 Problems/Problem 11
is on a semicircle with diameter and center Circle radius is tangent to and arc and circle radius is tangent to and arc . It is known that . The ratio can be expressed where are relatively prime positive integers. Find
Let the circle with radius have center and the circle with radius have center . Let the projections of and onto be and , respectively.
is equidistant from and , so it is on the angle bisector of .
We're given that , so . Now, we have , which is positive because .
We therefore also have . Now, , and we see that the radius of the large semicircle is .
Similarly, is the angle bisector of . Now, , and so , again positive because , and .
Now, . The radius of the large semicircle is thus .
Since the radii of the large semicircle are equal, we have , and so , and .