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cut in half
A rectangle.
If you cut a rectangle in half you wouldn't get a solid figure at all, since a rectangle is a plane figure. If you made a straight line cut you would get either a triangle or a quadrilateral of some variety depending on exactly how the cut was made.
No. There is no way you can cut one rectangle in half and only get one square. The other half would also have to be a square and this will only happen if the proportions are 1:2 and you cut the long sides.There are infinitely many ways to cut a rectangle in half none of which result in new rectangles. Only if the rectangle has proportions 1:2 can cutting it in half make a square, in fact the result could be two squares, not one.
Yes, you can cut a rectangle any way you please.