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HMMT 二月 2014 · 几何 · 第 1 题

HMMT February 2014 — Geometry — 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. Let O and O be concentric circles with radii 4 and 6, respectively. A chord AB is drawn in O with
    1 2 1
    length 2. Extend AB to intersect O in points C and D . Find CD . 2
解析

英文解析

  1. Let O and O be concentric circles with radii 4 and 6, respectively. A chord AB is drawn in O with
    1 2 1
    length 2. Extend AB to intersect O in points C and D . Find CD .
    √2
    Answer: 2 21 Let O be the common center of O and O , and let M be the midpoint of AB .
    1 2
    √ √
    2 2
    Then OM ⊥ AB , so by the Pythagorean Theorem, OM = 4 − 1 = 15. Thus CD = 2 CM =
    √ √
    2 6 − 15 = 2 21.2