HMMT 二月 2007 · 几何 · 第 9 题

英文解析

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 ?
   √
   Answer: 2 − 1 . Let AD = x , angle ABC = t . We also have ∠ BCA = 90 − t and ∠ DCA = 90 − 2 t 3
   so that ∠ ADC = 2 t . Considering triangles ABE and ADC , we obtain, respectively,
   cos( t ) = 1 / (1 + x ) and cos(2 t ) = x . By the double angle formula we get, (1 + x ) = 2.3
   Alternatively, construct M, the midpoint of segment BC, and note that triangles ABC, EBA, and
   M BD are similar. Thus, AB = BC · BE = BC. In particular,2
   BC AB BD 2 BD 2
   AB = = = = = ,
   AB BE BM BC AB2
   √ √
   3 3
   from which AB = 2 and AD = 2 − 1 .