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HMMT 二月 2007 · 几何 · 第 7 题

HMMT February 2007 — Geometry — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Convex quadrilateral ABCD has sides AB = BC = 7 , CD = 5 , and AD = 3. Given additionally ◦ that m ∠ ABC = 60 , find BD .
解析
  1. [ 5 ] Convex quadrilateral ABCD has sides AB = BC = 7 , CD = 5 , and AD = 3. Given additionally ◦ that m ∠ ABC = 60 , find BD . Answer: 8 . Triangle ABC is equilateral, so AC = 7 as well. Now the law of cosines shows that ◦ m ∠ CDA = 120 ; i.e., ABCD is cyclic. Ptolemy’s theorem now gives AC · BD = AB · CD + AD · BC , or simply BD = CD + AD = 8 .