HMMT 二月 2009 · 冲刺赛 · 第 27 题
HMMT February 2009 — Guts Round — Problem 27
题目详情
- [ 12 ] Circle Ω has radius 5. Points A and B lie on Ω such that chord AB has length 6. A unit circle ωis tangent to chord AB at point T. Given that ω is also internally tangent to Ω , find AT · BT.
12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth
英文原题
[ 12 ] Circle Ω has radius 5. Points A and B lie on Ω such that chord AB has length 6. A unit circle ω
is tangent to chord AB at point T. Given that ω is also internally tangent to Ω , find AT · BT.
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12 th HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUND
解析
英文解析
- [ 12 ] Circle Ω has radius 5. Points A and B lie on Ω such that chord AB has length 6. A unit circle ωis tangent to chord AB at point T. Given that ω is also internally tangent to Ω , find AT · BT.
Answer: 2
Solution: Let M be the midpoint of chord AB and let O be the center of Ω . Since AM = BM = 3 ,
Pythagoras on triangle AM O gives OM = 4 . Now let ω be centered at P and say that ω and Ω aretangent at Q. Because the diameter of ω exceeds 1, points P and Q lie on the same side of AB. Bytangency, O, P , and Q are collinear, so that OP = OQ − P Q = 4 . Let H be the orthogonal projectionof P onto OM ; then OH = OM − HM = OM − P T = 3 . Pythagoras on OHP gives HP = 7 . Finally,2
2 2 2 2
AT · BT = AM − M T = AM − HP = 9 − 7 = 2 .
12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth