2018 AMC 10A Problems/Problem 15
Contents
[hide]Problem
Two circles of radius are externally tangent to each other and are internally tangent to a circle of radius
at points
and
, as shown in the diagram. The distance
can be written in the form
, where
and
are relatively prime positive integers. What is
?
Solution 1
Let the center of the surrounding circle be
. The circle that is tangent at point
will have point
as the center. Similarly, the circle that is tangent at point
will have point
as the center. Connect
,
,
, and
. Now observe that
is similar to
. Writing out the ratios, we get
Therefore, our answer is
.
Solution 2
Let the center of the large circle be
. Let the common tangent of the two smaller circles be
. Draw the two radii of the large circle,
and
and the two radii of the smaller circles to point
. Draw ray
and
. This sets us up with similar triangles, which we can solve.
The length of
is equal to
by Pythagorean Theorem, the length of the hypotenuse is
, and the other leg is
. Using similar triangles,
is
, and therefore half of
is
. Doubling gives
, which results in
. Nice
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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