Mock AIME 1 2010 Problems/Problem 9
Problem
Let and be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle that is internally tangent to both and , and tangent to the line joining the centers of and . If the radius of can be expressed in the form , where , , and are integers, and is not divisible by the square if any prime, find the value of .
Solution
Let have center , have center , and have center . Further, let intersect at , at , and at , as in the diagram. Let be the radius of and let .
Because is tangent to , . Because and are tangent, we know that the line joining their centers goes through their point of tangency. Thus, because has radius , . Similarly, . Because with and , . Thus, , , and .
By the Pythagorean Theorem in , we have the following equation that we can solve for :
See Also
Mock AIME 1 2010 (Problems, Source) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |