HMMT February

Geometry Difficulty 5.3 AIME, harder Find the answer

Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is 3 feet tall and the tallest person is 7 feet tall, how tall is the middle person, in feet?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

If $A, B$ , and $C$ are the tops of the heads of three successive people and $D, E$ , and $F$ are their respective feet, let $P$ be the foot of the perpendicular from $A$ to $B E$ and let $Q$ be the foot of the perpendicular from $B$ to $C F$ . Then, by equal angles, $\triangle A B P \sim \triangle B C Q$ , so $\frac{C F}{B E}=\frac{C F}{B Q}=\frac{C Q}{B Q}+1=\frac{B P}{A P}+1=\frac{B E}{A P}=\frac{B E}{A D}$ Therefore the heights of successive people are in geometric progression. Hence, the heights of all five people are in geometric progression, so the middle height is $\sqrt{3 \cdot 7}=$ $\sqrt{21}$ feet.

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