Mock AIME I 2015 Problems/Problem 3
Problem 3
Let be points in the plane such that
,
, and
. Semicircles with diameters
and
intersect at a point
with
. Find the length of line segment
.
Solution
First note that is a 3-4-5 right triangle and
is a 20-21-29 right triangle. Thus we have
and
. Similarly, we have
and
. Applying the Law of Cosines on
gives us
We also know that
Hence, we have