Mock AIME 1 2010 Problems/Problem 9
Problem
Let and
be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle
that is internally tangent to both
and
, and tangent to the line joining the centers of
and
. If the radius of
can be expressed in the form
, where
,
, and
are integers, and
is not divisible by the square if any prime, find the value of
.
Solution
Let have center
,
have center
, and
have center
. Further, let
intersect
at
,
at
, and
at
, as in the diagram. Let
be the radius of
and let
.
Because is tangent to
,
. Because
and
are tangent, we know that the line joining their centers goes through their point of tangency. Thus, because
has radius
,
. Similarly,
. Because
with
and
,
. Thus,
,
, and
.
By the Pythagorean Theorem in , we have the following equation that we can solve for
:
, we have the following equation:
we found earlier, we see the following:
,
. Now, we can plug this value for
into our expression for
to get our answer:
.
See Also
Mock AIME 1 2010 (Problems, Source) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |