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PUMaC 2010 · 几何（A 组） · 第 6 题

PUMaC 2010 — Geometry (Division A) — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. In the following diagram, a semicircle is folded along a chord AN and intersects its diameter 2 M N at B . Given that M B : BN = 2 : 3 and M N = 10. If AN = x , find x .
解析
  1. In the following diagram, a semicircle is folded along a chord AN and intersects its diameter 2 M N at B . Given that M B : BN = 2 : 3 and M N = 10. If AN = x , find x . [Answer] 80 [Solution] Let C be symmetry point of B w.r.t. AN , then C is on arc AN with CN = BN = 6. √ 2 2 Then M C = 8. Suppose AN = x , then AM = 10 − x . Also, by symmetry, AM = AC . Apply Ptolemy’s Theorem on cyclic quadrilateral AM N C we get: AM · CN + M N · AC = AN · M C √ √ 2 2 2 2 ⇒ 6 10 − x + 10 10 − x = 8 x √ Solve for x we get x = 4 5.