HMMT 二月 2012 · TEAM2 赛 · 第 1 题
HMMT February 2012 — TEAM2 Round — Problem 1
题目详情
- [ 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
- [ 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 ? 5 Answer: Let E be the foot of the perpendicular from B to line AC . By the Law of Cosines, 3 4 cos ∠ BAC = , and it follows that BE = 3 and AE = 4. Now, by the Angle Bisector Theorem, 5 BD AB 5 5 = = , so BD = . BE AB + AE 9 3 n