2005 AMC 12A Problems/Problem 16
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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
Without loss of generality, let 
. 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 
and 
.
See also
| 2005 AMC 12A (Problems • Resources) | ||
| Preceded by Problem 15 | Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
