Maths Olympiad Prep

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Geometry Difficulty 4.3 AIME Find the answer United States

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon ABCDEFABCDEF, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at AA, BB, and CC are 1212, 99, and 1010 meters, respectively. What is the height, in meters, of the pillar at EE?

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Solution

Let MM be the midpoint of AC‾\overline{AC}. The pillars at AA and CC are 1212 and 1010 meters high, respectively, so the height of the panel at MM is 1111 meters. Because △BMA\triangle BMA is a 3030-6060-90∘90^\circ triangle, BMBM is half of the side length of the hexagon. Therefore BM=14BEBM = \frac{1}{4}BE. It follows that the height of the panel at point EE is 9+4⋅(11−9)=179 + 4 \cdot (11 - 9) = 17 meters.

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