2018 AMC 10A Problems/Problem 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 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
Call center of the largest circle . 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
which is choice
.
~Nivek