HMMT 二月 2011 · 冲刺赛 · 第 16 题
HMMT February 2011 — Guts Round — Problem 16
题目详情
- [ 10 ] Let R be a semicircle with diameter XY . A trapezoid ABCD in which AB is parallel to CD iscircumscribed about R such that AB contains XY . If AD = 4, CD = 5, and BC = 6, determine AB .
14 HARVARD-MIT MATHEMATICS TOURNAMENT, 12 FEBRUARY 2011 — GUTS ROUNDth
英文原题
[ 10 ] Let R be a semicircle with diameter XY . A trapezoid ABCD in which AB is parallel to CD is
circumscribed about R such that AB contains XY . If AD = 4, CD = 5, and BC = 6, determine AB .
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14 th HARVARD-MIT MATHEMATICS TOURNAMENT, 12 FEBRUARY 2011 — GUTS ROUND
解析
英文解析
- [ 10 ] Let R be a semicircle with diameter XY . A trapezoid ABCD in which AB is parallel to CD iscircumscribed about R such that AB contains XY . If AD = 4, CD = 5, and BC = 6, determine AB .
Answer: 10
We claim that AB = AD + BC . Let O denote the center of R . Since DA and DC are both tangentto R , we have ∠ ADO = ∠ ODC . Since CD is parallel to AB , we also have ∠ ODC = ∠ DOA . Thus
∠ ADO = ∠ DOA , and it follows that AD = AO . Similarly, we have BC = BO . Therefore, we
AB = AO + BO = AD + BC = 10 .