Kate is constructing a triangle. She knows that one side is 32 cm and another is 20 cm. What is the largest size for the last side of the triangle?

What is the smallest size for the last side of the triangle?

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.