2009/10 AHSMC Part 1 Problem 9

Problem 9 of 16

An equilateral triangle has area $2\sqrt{3}$ . From the midpoint of each side, perpendiculars are dropped to the other two sides. The area of the hexagon formed by these six lines is

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Solution

Let $ABC$ be the triangle and $DREPFQ$ be the hexagon, as shown in the diagram below.

Triangles $APE$ , $APF$ , $ERD$ and $FQD$ are all congruent to one another. Hence $DREPFQ$ has the same area as the parallelogram $AFDE$ , which is one half of $2\sqrt{3}$ . The answer is (C).

University of Alberta

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