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HMMT 二月 2007 · 几何 · 第 10 题

HMMT February 2007 — Geometry — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 8 ] ABCD is a convex quadrilateral such that AB = 2 , BC = 3 , CD = 7, and AD = 6. It also has anincircle. Given that ∠ ABC is right, determine the radius of this incircle. 1
解析

英文解析

  1. [ 8 ] ABCD is a convex quadrilateral such that AB = 2 , BC = 3 , CD = 7, and AD = 6. It also has anincircle. Given that ∠ ABC is right, determine the radius of this incircle.
    √
    1 + 13
    2 2 2 2 2
    Answer: . Note that AC = AB + BC = 13 = CD − DA . It follows that ∠ DAC is right,
    and so 3
    √ √
    [ ABCD ] = [ ABC ] + [ DAC ] = 2 · 3 / 2 + 6 · 13 / 2 = 3 + 3 13
    On the other hand, if I denotes the incenter and r denotes the inradius,
    [ ABCD ] = [ AIB ] + [ BIC ] + [ CID ] + [ DIA ] = AB · r/ 2 + BC · r/ 2 + CD · r/ 2 + DA · r/ 2 = 9 r
    √
    √
    1+ 13
    Therefore, r = (3 + 3 13) / 9 = .
    33