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HMMT 二月 2017 · 几何 · 第 4 题

HMMT February 2017 — Geometry — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Let ABCD be a convex quadrilateral with AB = 5 , BC = 6 , CD = 7, and DA = 8. Let M, P, N, Q be
    2 2
    the midpoints of sides AB, BC, CD, DA respectively. Compute M N − P Q .
解析

英文解析

  1. Let ABCD be a convex quadrilateral with AB = 5 , BC = 6 , CD = 7, and DA = 8. Let M, P, N, Q be
    2 2
    the midpoints of sides AB, BC, CD, DA respectively. Compute M N − P Q .
    Proposed by: Sam Korsky
    Answer: 13
    Draw in the diagonals of the quad and use the median formula three times to get M N in terms of the 2
    diagonals. Do the same for P Q and subtract, the diagonal length terms disappear and the answer is 2
    2 2 2 2
    BC + DA − AB − CD = 13 2