AMC8 2019

AMC8 2019 · Q6

AMC8 2019 · Q6. It mainly tests Probability (basic), Symmetry.

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?

图中显示的正方形中有 $81$ 个均匀分布的网格点，包括边缘上的点。点 $P$ 位于正方形的中心。从其他 $80$ 个点中随机选择点 $Q$，$PQ$ 线是正方形的对称轴的概率是多少？

(A) \frac{1}{5} \frac{1}{5}

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

(D) \frac{9}{20} \frac{9}{20}

(E) \frac{1}{2} \frac{1}{2}

Answer

Correct choice: (C)

正确答案：（C）

Solution

Lines of symmetry go through point $P$, and there are $8$ directions the lines could go, and there are $4$ dots at each direction.$\frac{4\times8}{80}=\boxed{\textbf{(C)} \frac{2}{5}}$.

对称轴通过点 $P$，有 $8$ 个方向，每方向有 $4$ 个点。$\frac{4\times8}{80}=\boxed{\textbf{(C)} \frac{2}{5}}$。

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