2001 AMC 12 Problems/Problem 18
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Problem
A circle centered at 
with a radius of 1 and a circle centered at 
with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
Solution
In the triangle 
we have 
and 
, thus by the Pythagorean theorem we have 
.
We can now pick a coordinate system where the common tangent is the 
axis and lies on the 
axis.
In this coordinate system we have 
and 
.
Let 
be the radius of the small circle, and let 
be the -coordinate of its center 
. We then know that 
, as the circle is tangent to the axis. Moreover, the small circle is tangent to both other circles, hence we have 
and 
.
We have 
and 
. Hence we get the following two equations:
Simplifying both, we get
As in our case both and are positive, we can divide the second one by the first one to get 
.
Now there are two possibilities: either 
, or 
. In the first case clearly 
, hence this is not the correct case. (Note: This case corresponds to the other circle that is tangent to both given circles and the axis - a large circle whose center is somewhere to the left of .) The second case solves to 
. We then have 
, hence 
.
See Also
| 2001 AMC 12 (Problems • Resources) | ||
| Preceded by Problem 17 | Followed by Problem 19 | |
| 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 | ||
