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HMMT 十一月 2011 · 冲刺赛 · 第 14 题

HMMT November 2011 — Guts Round — Problem 14

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

[8] Let AMOL be a quadrilateral with AM = 10, MO = 11, and OL= 12. Given that the perpen- dicular bisectors of sides AM andOLintersect at the midpoint of segment AO, find the length of side LA.

解析

英文解析

  1. [ 8 ] Let AM OL be a quadrilateral with AM = 10, M O = 11, and OL = 12. Given that the perpendicular bisectors of sides AM and OL intersect at the midpoint of segment AO , find the length of side
    LA .
    √
    Answer: 77 Let D be the midpoint of AM and E be the midpoint of AO . Then, we note that
    ADE ∼ AM O , so M is a right angle. Similarly, L is a right angle. Consequently, we get that
    √ √
    √
    2 2 2
    2 2 2 2 2
    AO = OM + AM ⇒ AL = AO − OL = 11 + 10 − 12 = 77 .
    Guts Round
    A OEDM L