There are 81 grid points (uniformly spaced). 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 the line of symmetry for the square?
2019 AMC 8 Question 6
A square has different lines of symmetry. For example, a vertical line through $P$ is one such line of symmetry (because then the square is the same to the left and to the right of the line):
Note this line contains $P$ and $4+4 = 8$ other points. Therefore, if $Q$ is one of these $8$ points, $PQ$ is a line of symmetry for the square.
With one line of symmetry done, how many other lines of symmetry do we need to consider? How many other points do they contain?