# 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 .

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