2005 AMC 12A Problems/Problem 16
Problem
Three circles of radius are drawn in the first quadrant of the -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the -axis, and the third is tangent to the first circle and the -axis. A circle of radius is tangent to both axes and to the second and third circles. What is ?
Solution
Solution 1
Set so that we only have to find . Draw the segment between the center of the third circle and the large circle; this has length . We then draw the radius of the large circle that is perpendicular to the x-axis, and draw the perpendicular from this radius to the center of the third circle. This gives us a right triangle with legs and hypotenuse . The Pythagorean Theorem yields:
Quite obviously , so .
Solution 2
Applying Wildin's Theorem directly yields
As before, , so .
(Note that Wildin's Theorem is not a valid theorem for many AMC 12 problems.)
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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