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HMMT 二月 2008 · 冲刺赛 · 第 2 题

HMMT February 2008 — Guts Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] Given right triangle ABC , with AB = 4 , BC = 3 , and CA = 5. Circle ω passes through A and istangent to BC at C . What is the radius of ω ?
解析

英文解析

  1. [ 5 ] Given right triangle ABC , with AB = 4 , BC = 3 , and CA = 5. Circle ω passes through A and istangent to BC at C . What is the radius of ω ?
    Answer: Let O be the center of ω , and let M be the midpoint of AC . Since OA = OC ,25
    OM ⊥ AC . Also, ∠ OCM = ∠ BAC , and so triangles ABC and CM O are similar. Then, CO/CM =8
    AC/AB , from which we obtain that the radius of ω is CO = .25 8