AMC 8 2010 Problem 17

AMC8 2010 年第 17 题

面积与周长（综合）★★★☆☆
The diagram shows an octagon consisting of 10 unit squares. The portion below PQ‾\overline{PQ} is a unit square and a triangle with base 5 . If PQ‾\overline{PQ} bisects the area of the octagon, what is the ratio XQQY\displaystyle \frac{XQ}{QY}?
下图是一个由 10 个单位正方形组成的八边形。PQ‾\overline{PQ} 下方的部分是一个单位正方形和一个底边长为 5 的三角形。若 PQ‾\overline{PQ} 平分八边形的面积，那么 XQQY\displaystyle \frac{XQ}{QY} 的比值是多少？
AMC 8 2010 Problem 17 figure
  1. A.25\displaystyle \frac 25
  2. B.12\displaystyle \frac 12
  3. C.35\displaystyle \frac 35
  4. D.23\displaystyle \frac 23
  5. E.34\displaystyle \frac 34

正确答案 · Correct Answer

D

解析 · Solution

Solution 1

We see that half the area of the octagon is 5. We see that the triangle area is 5-1 = 4. That means that 5h2=4→h=85\displaystyle \frac{5h}{2} = 4 \rightarrow h = \frac{8}{5}. QY=85−1=35\displaystyle \text{QY} = \frac{8}{5} - 1 = \frac{3}{5} Meaning, 2535=23\displaystyle \frac{\frac{2}{5}}{\frac{3}{5}} = \frac{2}{3}

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