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HMMT 二月 2006 · 几何 · 第 8 题

HMMT February 2006 — Geometry — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Triangle ABC has a right angle at B . Point D lies on side BC such that 3 ∠ BAD = ∠ BAC . Given AC = 2 and CD = 1, compute BD .
解析
  1. Triangle ABC has a right angle at B . Point D lies on side BC such that 3 ∠ BAD = ∠ BAC . Given AC = 2 and CD = 1, compute BD . 3 Answer: 8 ′ ′ Solution: Let BD = x . We reflect D over AB to D . Then DD = 2 x , but AD √ √ ′ ′ 2 2 2 2 2 bisects CAD , so 4 x = AD = AD . Also, AD = x + AB = x + AC − BC = √ √ 2 2 2 x + 4 − ( x + 1) = 3 − 2 x . We have the quadratic 16 x = 3 − 2 x which gives x = 3 / 8.