Find all ways to cut a rectangle into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different dimensions (side lengths) than the orgininal one.

I already know countably many ways.

update: infinitely many.

Does “infintely many” mean “uncountably many”?

No, both mean countably and infinitely many. I updated because “countably” often means “not uncountable”, i.e. includes “finitely”