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HMMT 二月 2003 · 冲刺赛 · 第 33 题

HMMT February 2003 — Guts Round — Problem 33

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [10] We are given triangle ABC , with AB = 9, AC = 10, and BC = 12, and a point
    ′ ′
    D on BC . B and C are reflected in AD to B and C , respectively. Suppose that lines
    ′ ′
    BC and B C never meet (i.e., are parallel and distinct). Find BD .
    HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND
解析

英文解析

  1. We are given triangle ABC , with AB = 9, AC = 10, and BC = 12, and a point D on
    ′ ′ ′
    BC . B and C are reflected in AD to B and C , respectively. Suppose that lines BC
    ′
    and B C never meet (i.e., are parallel and distinct). Find BD .
    Solution: 6
    The lengths of AB and AC are irrelevant. Because the figure is symmetric about AD ,
    ′ ′
    lines BC and B C meet if and only if they meet at a point on line AD . So, if they
    ′
    never meet, they must be parallel to AD . Because AD and BC are parallel, triangles
    ′
    ABD and ADC have the same area. Then ABD and ADC also have the same area.
    Hence, BD and CD must have the same length, so BD = BC = 6.1 2