Pythagorean theorem

Question

We can make a square into four equal squares. Fine, if we want to make into five.. Then there is a problem. Please discuss, How to make five squares from a single square by using a Pythagorean theorem. Is there any other way to make five squares from one square without using Pythagorean theorem? Please discuss. Thanking you, KKRG

Answer 1

Five equal squares can be cut into a total of nine pieces that can be reassembled to form a single square. See this Wolfram demonstration.

EDIT: You cut four of the (unit) squares into two pieces each, and in the same way: you cut along a line from a corner to the midpoint of a side. The two pieces of each of these squares can be put back together to form a right triangle with sides 1 and 2, and hypotenuse $\sqrt5$. The four triangles can then be placed around the remaining square to form the big square with side $\sqrt5$.

If you want to see it in reverse, start with the big square, and cut from each corner to the midpoint of a side; Northwest corner to South side, Northeast corner to West side, SE to N, and SW to E. That cuts the big square in 9 pieces. One of the 9 pieces is a square. The other 8 are 4 little triangles and 4 trapezoids. Each triangle can be fitted to a trapezoid to form another small square.