HMMT 二月 2005 · 冲刺赛 · 第 7 题
HMMT February 2005 — Guts Round — Problem 7
题目详情
- [6] 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?
解析
- 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? √ Solution: 21 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 BE and let Q be the foot of the perpendicular from B to CF . Then, by equal angles, 4 ABP ∼ 4 BCQ , so CF CF CQ BP BE BE = = + 1 = + 1 = = . BE BQ BQ AP AP AD 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 3 · 7 = √ 21 feet. 2 C B Q A P D E F