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**Teacher Information: Pythagorean Puzzles**

This virtual manipulative gives two dynamic illustrations of the Pythagorean Theorem, that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. These proofs by dissection are only two of literally hundreds of ways that people have found to demonstrate the validity of the theorem. Students who have played with rearrangements of shaped pieces may solve the puzzles rather quickly; those without any experience may need patience and perhaps some hints.

In Puzzle #1, the red triangles and blue square are all congruent, as may be seen by dragging one on top of another. The challenge is to arrange the pieces to exactly fill the identical white square regions (the key being to rotate the blue square on the left appropriately). In looking for a starting point for trying to fit the red triangles into the white square, the only possible way to fit a side of a triangle to fill up a side of the square is to use the hypotenuse of the triangle on the side of the square, so some rotation is necessary. Similarly, to start on the odd-shaped white region on the right, which side of the triangle must go on the vertical edge on the left side?

In Puzzle #2, after a student has worked with Puzzle #1, the arrangements should be fairly straightforward, but if hints are needed, the triangles surround the green square in the left region, and two red triangles fit together to make rectangles which can be used to fill the right region. After that, the argument for the validity of the Pythagorean Theorem is more easily seen by removing the four congruent triangles from each figure. Since the removed triangles are identical in both sets, the remaining pieces must also have equal areas, so that the green square (with area //c//2) must have the same area as the two blue squares (//a//2 +//b//2).