To the left is an animated proof of the Pythagorean Theorem. Starting with a
right triangle and squares on each side, the middle size square is cut
into congruent quadrilaterals (the cuts through the center and
parallel to the sides of the biggest square). Then the quadrilaterals
are hinged and rotated and shifted to the big square. Finally the
smallest square is translated to cover the remaining middle part of
the biggest square. A perfect fit! Thus the sum of the squares on the
smaller two sides equals the square on the biggest side.
What this means in plain language:
As you know, the area of a
square is the length of its side, multiplied by itsemf (squared). For
instance, the area of a square room that is 10 by 10 feet is 10
multiplied by 10, that is, 100 square feet.
What this animated proof says is this: take a square whose side is
the same as the hypothenuse (green square). You can take scissors and cut this
square into pieces that can be REASSEMBLED to become two
squares. These two squares would be with the sides of the same length
as the sides of the right triangle.
Because cutting the big square with scissors and reassembling it into smaller pieces did not change the total area of the piece of paper, we now know that the area of the big square is the sum of areas of the small squares, or 
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