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HMMT 二月 2011 · 冲刺赛 · 第 1 题

HMMT February 2011 — Guts 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. [ 4 ] Let ABC be a triangle with area 1. Let points D and E lie on AB and AC , respectively, suchthat DE is parallel to BC and DE/BC = 1 / 3. If F is the reflection of A across DE , find the area oftriangle F BC .
解析

英文解析

  1. [ 4 ] Let ABC be a triangle with area 1. Let points D and E lie on AB and AC , respectively, suchthat DE is parallel to BC and DE/BC = 1 / 3. If F is the reflection of A across DE , find the area oftriangle F BC .
    Answer:1
    Let AF intersect BC at H . Since DE/BC = 1 / 3 and F and A are equidistant from DE , we have 3
    2 1
    AF = AH and F H = AH − AF = AH . Furthermore, since AF is perpendicular to DE , we have
    3 3
    AH and F H are the altitudes of triangles ABC and F BC respectively. Therefore the area of triangle
    1 1 1 1
    F BC is · F H · BC = · · AH · BC = .
    2 2 3 3