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Ratio of segments divided by orthocenter

Source: AHSME 1964 #35

June 15, 2012
ratiogeometrytrigonometryEulercircumcirclearea of a triangleHeron's formula

Problem Statement

The sides of a triangle are of lengths 1313, 1414, and 1515. The altitudes of the triangle meet at point HH. If ADAD is the altitude to the side length 1414, what is the ratio HD:HAHD:HA?
<spanclass=′latex−bold′>(A)</span>3:11<spanclass=′latex−bold′>(B)</span>5:11<spanclass=′latex−bold′>(C)</span>1:2<spanclass=′latex−bold′>(D)</span>2:3<spanclass=′latex−bold′>(E)</span>25:33<span class='latex-bold'>(A) </span> 3 : 11\qquad <span class='latex-bold'>(B) </span> 5 : 11\qquad <span class='latex-bold'>(C) </span> 1 : 2\qquad <span class='latex-bold'>(D) </span>2 : 3\qquad <span class='latex-bold'>(E) </span>25 : 33
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