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HMMT 二月 2012 · TEAM2 赛 · 第 1 题

HMMT February 2012 — TEAM2 Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] Triangle ABC has AB = 5, BC = 3 2, and AC = 1. If the altitude from B to AC and the angle bisector of angle A intersect at at D , what is BD ? n
解析

英文解析

  1. [ 10 ] Triangle ABC has AB = 5, BC = 3 2, and AC = 1. If the altitude from B to AC and the anglebisector of angle A intersect at at D , what is BD ?
    Answer: Let E be the foot of the perpendicular from B to line AC . By the Law of Cosines,5
    cos ∠ BAC = , and it follows that BE = 3 and AE = 4. Now, by the Angle Bisector Theorem,43
    BD AB 5 55 = = , so BD = .
    BE AB + AE 9 3 n