Question

下图的正方形中有 $81$ 个均匀分布的格点，包括边上的点。点 $P$ 在正方形中心。若点 $Q$ 从其余 $80$ 个点中随机选取，那么直线 $PQ$ 是正方形对称轴的概率是多少？

There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?

$\displaystyle \dfrac{1}{5}$

$\displaystyle \dfrac{1}{4}$

$\displaystyle \dfrac{2}{5}$

$\displaystyle \dfrac{9}{20}$

$\displaystyle \dfrac{1}{2}$

答案：C

知识点：基本概率对称性

难度评级：1070

视频讲解：

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文字解答：

正方形的对称轴只有两条对角线和两条连接对边中点的直线。

因此 $Q$ 必须落在这四条线中的一条上。每条线上有 $8$ 个格点，四条线共计 $4 \cdot 8 = 32$ 个点。但不能和 $P$ 重合，而中心点 $P$ 在四条线上被重复计入了次，所以可选点数为。

概率为 $\dfrac{32}{80} = \dfrac{2}{5}$ 。

正确答案是 C。

Note the only lines of symmetry for a square are the two diagonals and the two lines connecting opposite midpoints.

On each symmetry line there are $8$ grid points other than $P.$ The four lines intersect only at $P,$ which is not eligible, so these give $4 \cdot 8 = 32$ distinct choices for $Q.$

The probability is therefore $\dfrac{32}{80} = \dfrac{2}{5}.$

Thus, the correct answer is C.

2019 AMC 8 第 6 题

答案：C

难度评级：1070

视频讲解：

文字解答：

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