AMC 8 2013 Problem 23

AMC8 2013 年第 23 题

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Angle ABC of △ABC\triangle ABC is a right angle. The sides of △ABC\triangle ABC are the diameters of semicircles as shown. The area of the semicircle on AB‾\overline{AB} equals 8π8\pi, and the arc of the semicircle on AC‾\overline{AC} has length 8.5π8.5\pi. What is the radius of the semicircle on BC‾\overline{BC}?
在内，∠ABC 是一个直角。 的三条边是如图所示的三个半圆的直径。 上的半圆面积是 8π，边上的半圆弧长为 8.5π．那么边上的半圆的半径是多少？
边
AMC 8 2013 Problem 23 figure
  1. A.7
  2. B.7.5
  3. C.8
  4. D.8.5
  5. E.9

正确答案 · Correct Answer

B

解析 · Solution

解法1

常规解法 · 还原直径 + 勾股 AB‾\overline{AB} 上的半圆若补成整圆，面积 16π16\pi，由 πr2=16π\pi r^2=16\pi 得 r=4，直径 AB=8。AC‾\overline{AC} 上半圆弧长 πr=8.5π\pi r=8.5\pi，得 r=8.5，直径 AC=17。直角在 BB，由勾股 BC=172−82=15BC=\sqrt{17^2-8^2}=15，半圆半径 7.5。

Solution 1

Elementary · recover the diameters + Pythagoras If the semicircle on AB‾\overline{AB} were a full circle its area would be 16π16\pi, so r=4 and AB=8. The semicircular arc on AC‾\overline{AC} has length πr=8.5π\pi r=8.5\pi, so r=8.5 and AC=17. Since the right angle is at BB, BC=172−82=15BC=\sqrt{17^2-8^2}=15, and the semicircle on BC‾\overline{BC} has radius 7.5.

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