AMC 8 2019 Problem 6

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?

$\textbf{(A) }\frac{1}{5} \textbf{(B) }\frac{1}{4} \textbf{(C) }\frac{2}{5} \textbf{(D) }\frac{9}{20} \textbf{(E) }\frac{1}{2}$