2014 AIME II Problems/Problem 14
PROBLEM
In
, and
. Let
and
be points on the line
such that
,
, and
. Point
is the midpoint of the segment
, and point
is on ray
such that
. Then
, where
and
are relatively prime positive integers. Find
.
DIAGRAM
http://www.artofproblemsolving.com/Wiki/images/5/59/AOPS_wiki.PNG ( This is the diagram.)
SOLUTION
As we can see,
is the midpoint of
and
is the midpoint of
is a
triangle, so
.
is
triangle.
and
are parallel lines so
is
triangle also.
Then if we use those informations we get and
and
or
Now we know that , we can find for
which is simpler to find.
We can use point to split it up as
,
We can chase those lengths and we would get
, so
, so
, so
Then using right triangle , we have HB=10 sin (15∘)
So HB=10 sin (15∘)=.
And we know that .
Finally if we calculate .
. So our final answer is
.
Thank you.
-Gamjawon