AMC10 2011 B

AMC10 2011 B · Q16

AMC10 2011 B · Q16. It mainly tests Probability (basic), Area & perimeter.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

飞镖盘是一个正八边形，按照图示划分成区域。假设飞镖投向飞镖盘时，均匀随机落在飞镖盘的任何位置。飞镖落在中心正方形内的概率是多少？

(A) $\frac{\sqrt{2}-1}{2}$ $\frac{\sqrt{2}-1}{2}$

(B) $\frac{1}{4}$ $\frac{1}{4}$

(D) $\frac{\sqrt{2}}{4}$ $\frac{\sqrt{2}}{4}$

(E) $2-\sqrt{2}$ $2-\sqrt{2}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Assume the octagon’s edge is 1. Then the corner triangles have hypotenuse 1 and thus legs $\frac{\sqrt{2}}{2}$ and area $\frac{1}{4}$ each; the four rectangles are $1$ by $\frac{\sqrt{2}}{2}$ and have area $\frac{\sqrt{2}}{2}$ each, and the center square has area $1$. The total area is $4\cdot\frac{1}{4}+4\cdot\frac{\sqrt{2}}{2}+1=2+2\sqrt{2}$. The probability that the dart hits the center square is $\frac{1}{2+2\sqrt{2}}=\frac{\sqrt{2}-1}{2}$.

答案（A）：设八边形的边长为 $1$。则四个角上的三角形斜边为 $1$，因此两直角边为 $\frac{\sqrt{2}}{2}$，每个面积为 $\frac{1}{4}$；四个长方形的尺寸为 $1\times\frac{\sqrt{2}}{2}$，每个面积为 $\frac{\sqrt{2}}{2}$；中心正方形面积为 $1$。总面积为 $4\cdot\frac{1}{4}+4\cdot\frac{\sqrt{2}}{2}+1=2+2\sqrt{2}$。飞镖落在中心正方形内的概率为 $\frac{1}{2+2\sqrt{2}}=\frac{\sqrt{2}-1}{2}$。

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