HMMT 二月 2012 · 几何 · 第 2 题
HMMT February 2012 — Geometry — Problem 2
题目详情
英文原题
Let ABC be a triangle with ∠A= 90◦,AB= 1, and AC= 2. Let ℓbe a line through Aperpendicular
toBC, and let the perpendicular bisectors of ABandACmeet ℓatEandF, respectively. Find the
length of segment EF.
解析
英文解析
- Let ABC be a triangle with ∠ A = 90 , AB = 1, and AC = 2. Let ℓ be a line through A perpendicularto 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
√
∠ ACB and ∠ EAC = ∠ ABC , so AEM ∼ CBA ⇒ AE = and F AN ∼ CBA ⇒ AF = 5.5
√4
3 5
Consequently, EF = AF − AE = . 4