If you cut a square diagonally from corner to corner you get four right-angled isosceles triangles.
You may only fit long sides to long sides and short sides to short sides.
You could use this interactivity to try out your ideas:
Click on the grey triangles to make new pieces.
You can rotate them by dragging the corners.
Printable NRICH Roadshow resource.
You may be interested in the other problems in our I Wonder... What If...? Feature.