HMMT 二月 2005 · 冲刺赛 · 第 7 题
HMMT February 2005 — Guts Round — Problem 7
题目详情
英文原题
- [6] Five people of different heights are standing in line from shortest to tallest. As ithappens, 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 theshortest person is 3 feet tall and the tallest person is 7 feet tall, how tall is the middleperson, in feet?
解析
英文解析
- Five people of different heights are standing in line from shortest to tallest. As ithappens, 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 theshortest person is 3 feet tall and the tallest person is 7 feet tall, how tall is the middleperson, in feet?
√
Solution: 21
If A , B , and C are the tops of the heads of three successive people and D , E , and F aretheir respective feet, let P be the foot of the perpendicular from A to BE and let Q bethe foot of the perpendicular from B to CF . Then, by equal angles, 4 ABP ∼ 4 BCQ ,
CF CF CQ BP BE BEso = = + 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.
C2
B Q
A P
D E F