Set of regular polygons

Question

Defined set S of regular polygons on plane.

Known for every polygon: at least one of it's sides is parallel to x,and this side's length is rational number.

Also known- there is at least one vertex on this side that it's coordinates are rational numbers.

what is the cardinality of S?

thanks.

Answer 1

HINT: Let $P$ be a polygon in $S$, and let $\sigma$ be a side of $P$ parallel to the $x$-axis, let $p$ be an endpoint of $\sigma$ with rational coordinates, and let $n$ be the number of sides that $P$ has. Then $P$ is completely determined by the following data:

• the coordinates of $p$;
• whether $p$ is the left or right endpoint of $\sigma$;
• the length of $\sigma$;
• whether $\sigma$ is the top or the bottom edge of $P$; and
• $n$.

How many different combinations of values are possible?