HMMT 二月 2007 · 几何 · 第 9 题

HMMT February 2007 — Geometry — Problem 9

[ 7 ] 4 ABC is right angled at A . D is a point on AB such that CD = 1. AE is the altitude from A to BC . If BD = BE = 1, what is the length of AD ?

解析

[ 7 ] 4 ABC is right angled at A . D is a point on AB such that CD = 1. AE is the altitude from A to BC . If BD = BE = 1, what is the length of AD ? √ 3 Answer: 2 − 1 . Let AD = x , angle ABC = t . We also have ∠ BCA = 90 − t and ∠ DCA = 90 − 2 t so that ∠ ADC = 2 t . Considering triangles ABE and ADC , we obtain, respectively, 3 cos( t ) = 1 / (1 + x ) and cos(2 t ) = x . By the double angle formula we get, (1 + x ) = 2. Alternatively, construct M, the midpoint of segment BC, and note that triangles ABC, EBA, and 2 M BD are similar. Thus, AB = BC · BE = BC. In particular, BC AB BD 2 BD 2 AB = = = = = , 2 AB BE BM BC AB √ √ 3 3 from which AB = 2 and AD = 2 − 1 .