1997 AIME Problems/Problem 4
Problem
Circles of radii and are mutually externally tangent, where and are relatively prime positive integers. Find
Solution
If (in the diagram above) we draw the line going through the centers of the circles with radii and , that line is the perpendicular bisector of the segment connecting the centers of the two circles with radii . Then we form two right triangles, of lengths and , wher is the distance between the center of the circle in question and the segment connecting the centers of the two circles of radii . By the Pythagorean Theorem, we now have two equations with two unknowns:
So .
NOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been formed.
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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