2014 AIME II Problems/Problem 14
14. In , and . Let and be points on the line such that AH⊥BC, , and . Point is the midpoint of the segment , and point is on ray such that PN⊥BC. Then , where and are relatively prime positive integers. Find .
http://www.artofproblemsolving.com/Wiki/images/5/59/AOPS_wiki.PNG ( This is the diagram.)
As we can see,
is the midpoint of and is the midpoint of
is a triangle, so .
and are parallel lines so is triangle also.
Then if we use those informations we get and
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 .