PUMaC 2014 · 几何（A 组） · 第 7 题

PUMaC 2014 — Geometry (Division A) — Problem 7

[ 7 ] Let O be the center of a circle of radius 26, and let A, B be two distinct point on the circle, o with M being the midpoint of AB . Consider point C for which CO = 34 and ∠ COM = 15 . o Let N be the midpoint of CO . Suppose that ∠ ACB = 90 . Find M N .

解析

[ 7 ] Let O be the center of a circle of radius 26, and let A, B be two distinct point on the circle, o with M being the midpoint of AB . Consider point C for which CO = 34 and ∠ COM = 15 . o Let N be the midpoint of CO . Suppose that ∠ ACB = 90 . Find M N . Solutiion: We apply cosine rule to 4 M N O and 4 M N C , to get 2 2 2 2 2 OM = ON + M N − 2 M N · ON cos ∠ M N O = 17 + M N − 34 M N cos ∠ M N O 2 2 2 2 2 CM = N C + M N + 2 M N · N C cos ∠ M N O = 17 + M N + 34 M N cos ∠ M N O 2 2 2 2 2 2 Hence we have OM + CM = 2 × 17 + 2 × M N . Since CM = AM Hence OM + CM = √ 2 2 2 OP = 26 Hence M N = 13 × 26 − 17 = 7 2