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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

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What have you tried so far? Please post. –  draks ... Jun 26 '12 at 13:56
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What operations are allowed? Are you cutting the large square into pieces that need to be rearranged to form five smaller squares? Do all the smaller squares need to be the same size? What do you mean without using the Pythagorean theorem? I can know 9+16=25 even without Pythagoras. –  Ross Millikan Jun 26 '12 at 13:59
    
We can make a square into four equal squares., but $9=4\cdot 1.25$, so what do you mean? –  draks ... Jun 26 '12 at 14:04
    
You can cut a big square into 5 equal squares in size and area by using Pythagorean theorem. You are allowed for any operations. the 5 pieces when you add, we should get a big square. –  KRRG-BITS Jun 26 '12 at 15:13
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@KRRG: We're allowed any operations? Then I choose the operation of creating 4 identical duplicates of the original square. –  Henning Makholm Jun 26 '12 at 16:29

1 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.

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! your pictures look like a rectangle. You should explain at what length you cut and pasted. So that, any one can understand about the figure square or rectangle. But, the animation you made is very very good. Thank you. –  KRRG-BITS Jun 27 '12 at 3:25
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I didn't make anything: I found it at that URL. The picture shows 5 squares. That they start out next to each other (I assume this is what you mean when you say they "look like a rectangle") is supposed to make it easy for you to see exactly where and how the cuts are made. Try to work out the details on your own, remembering that we know that if the little squares have side 1 then the big square has side $\sqrt5$. If you get stuck, maybe I'll have a look at the demo, or maybe someone else will, and see what we can figure out. –  Gerry Myerson Jun 27 '12 at 4:02
    
! why I am giving reply late you know. I am really trying to prove mathematically by using Pythagorean theorem. But, I was failed to get. My intention that, can we make five squares from 1 square? as per your wolfram demo, I understand we can. But, can we prove mathematically...? from my side, I given several attempts. Still I could not. If possible, please let me know from your side. Please... –  KRRG-BITS Jun 27 '12 at 16:58
    
I have put some more detail in my answer. I hope it helps. –  Gerry Myerson Jun 28 '12 at 0:44

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