HMMT 二月 2014 · 几何 · 第 4 题
HMMT February 2014 — Geometry — Problem 4
题目详情
英文原题
- In quadrilateral ABCD , \ DAC = 98 , \ DBC = 82 , \ BCD = 70 , and BC = AD . Find \ ACD .
82AB
C D98
解析
英文解析
- In quadrilateral ABCD , ∠ DAC = 98 , ∠ DBC = 82 , ∠ BCD = 70 , and BC = AD . Find ∠ ACD .
°AB
°82
C D98
′ ′
Answer: 28 Let B be the reflection of B across CD . Note that AD = BC , and ∠ DAC + ∠ CB D =
° ′ ′ ′ °
180 , so ACB D is a cyclic trapezoid. Thus, ACB D is an isosceles trapezoid, so ∠ ACB = 98 . Note
′ ° ′ ′ ° ° °
that ∠ DCB = ∠ BCD = 70 , so ∠ ACD = ∠ ACB − ∠ DCB = 98 − 70 = 28 .