Difference between revisions of "2014 AIME II Problems/Problem 14"
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Then if we use those informations we get <math>AD=2HD</math> and | Then if we use those informations we get <math>AD=2HD</math> and | ||
− | <math>PD=2ND</math> and <math>AP= | + | <math>PD=2ND</math> and <math>AP=AD-PD=2HD-2ND=2HN</math> or <math>AP=2HN=HM</math> |
Revision as of 21:35, 29 March 2014
14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line BC such that AH⊥BC, ∠BAD=∠CAD, 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 ∠HAB=15∘.
is .
and are parallel lines so is also.
Then if we use those informations we get and
and or