There are 81 grid points ( uniformly spaced ) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that the line $PQ$ is a line of symmetry for the square?
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $\frac{2}{5}$
D. $\frac{9}{20}$
E. $\frac{1}{2}$

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Worked solution

1. 2019 AMC 8 官方解析

在一个均匀分布的正方形区域，边上和内部共有81个格点．点是正方形的中心．若是其余80个点中随机抽取的一个，那么直线是正方形对称轴的概率是？

There are four lines of symmetry of the square and all pass through $P$ . Those four are made up with: two diagonals, and two lines joining mid points of opposite sides. Count all such $Q$
:on each half-line.
$\frac{4}{80} \times 8 = \frac{2}{5}$ .

Answer

C. $\frac{2}{5}$

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