Draw a right triangle (shown in orange above) with squares on its sides, shown in white, green and blue above. We are going to show that the green and blue squares on the triangle’s two legs can be cut up and fit into the white square on the triangle’s hypotenuse. So we are going to show that the sum of the squares on the two legs of a right triangle equals the square on the hypotenuse.

Subdivide the square on the larger leg, the blue square, as shown.

Now cut the big blue square on the on the longer leg into four pieces along the dotted lines as shown, and reassemble the four pieces and the smaller green square to cover the white square on the hypotenuse. (The four pieces can just be slid into place; they do not need to be rotated or flipped.)

Notice that the four vertices of the new blue and green square built on the hypotenuse are exactly the same four vertices that you see where the two perpendicular lines meet in the blue square on the leg of the triangle above!

Do you see it?

So the sum of the squares on the legs of a right triangle equals the square on the hypotenuse.