Two hikers started at the same location. One traveled 2 miles east and then 1 mile north. The
other traveled 1 mile west and then 3 miles south.
At the end of their hikes, how many miles apart are
the two hikers? [The use of the accompanying grid
is optional.]

Set $r$ is a set of rectangles such that (1) only the grid points shown here are used as vertices, (2) all sides are vertical or horizontal and (3) no two rectangles in the set are congruent. if $r$ contains the maximum possible number of rectangles given these conditions, what fraction of the rectangles in set $r$ are squares? express your answer as a common fraction.