返回题库

HMMT 二月 2012 · 几何 · 第 2 题

HMMT February 2012 — Geometry — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let ABC be a triangle with ∠ A = 90 , AB = 1, and AC = 2. Let ℓ be a line through A perpendicular to BC , and let the perpendicular bisectors of AB and AC meet ℓ at E and F , respectively. Find the length of segment EF .
解析
  1. Let ABC be a triangle with ∠ A = 90 , AB = 1, and AC = 2. Let ℓ be a line through A perpendicular to BC , and let the perpendicular bisectors of AB and AC meet ℓ at E and F , respectively. Find the length of segment EF . √ 3 5 Answer: Let M, N be the midpoints of AB and AC , respectively. Then we have ∠ EAB = 4 √ √ 5 ∠ ACB and ∠ EAC = ∠ ABC , so AEM ∼ CBA ⇒ AE = and F AN ∼ CBA ⇒ AF = 5. 4 √ 3 5 Consequently, EF = AF − AE = . 4