Question

如图，一个飞镖盘是被分成若干区域的正八边形。假设飞镖落在盘内任一点的可能性相同。飞镖落在中心正方形内的概率是多少？

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?

$\dfrac{\sqrt{2}-1}{2}$

$\dfrac{1}{4}$

$\dfrac{2-\sqrt{2}}{2}$

$\dfrac{\sqrt{2}}{4}$

$2-\sqrt{2}$

答案：A

知识点：几何概率面积分割正多边形

难度评级：1480

小提示：

设八边形边长为 $1$ ，把它分成中心正方形、四个长方形和四个角上的三角形。

Set the octagon’s edge to $1$ and split it into the center square, four rectangles, and four corner triangles

大提示：

角上的三角形是直角等腰三角形，直角边长为 $\dfrac{\sqrt2}{2}$ 。

The corner triangles are right isosceles with legs $\dfrac{\sqrt2}{2}$

解答：

设八边形边长为 $1$ 。四个角上的三角形是直角等腰三角形，直角边长为 $\dfrac{\sqrt2}{2}$ ，每个面积为 $\dfrac14$ 。四个长方形尺寸为 $1$ 乘 $\dfrac{\sqrt2}{2}$ ，每个面积为 $\dfrac{\sqrt2}{2}$ ，中心正方形面积为 $1$ 。

总面积为 $4\cdot\dfrac14+4\cdot\dfrac{\sqrt2}{2}+1=2+2\sqrt2\text{。}$ 因此击中中心正方形的概率为 $\dfrac{1}{2+2\sqrt2}=\dfrac{\sqrt2-1}{2}\text{。}$

所以正确答案是 A。

Assume the octagon has edge length $1.$ The four corner triangles are right isosceles with legs $\dfrac{\sqrt2}{2}$ and area $\dfrac14$ each. The four rectangles are $1$ by $\dfrac{\sqrt2}{2}$ with area $\dfrac{\sqrt2}{2}$ each, and the center square has area $1.$

The total area is $4\cdot\dfrac14+4\cdot\dfrac{\sqrt2}{2}+1=2+2\sqrt2.$ The probability of hitting the center square is $\dfrac{1}{2+2\sqrt2}=\dfrac{\sqrt2-1}{2}.$

Thus, the correct answer is A.

2011 AMC 12B 第 12 题

答案：A

难度评级：1480

小提示：

大提示：

解答：

其他年份的第 12 题