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Proof Without Words: Pythagorean Theorem

Math Content:

Why is c2 = a2 + b2? Watch a dynamic, geometric "proof without words" of the Pythagorean Theorem. Can you explain the proof?

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  • The sliders at the top of the screen indicate the lengths (a and b) of the legs of the right triangle. Adjust these sliders to change the size of the triangle.
  • Press the Arrange Four Copies button to begin the "proof without words." More buttons will appear to lead you through the proof.
  • The Start Over button will return you to the beginning.

This proof is attributed to Bhaskara (12th century) in the book Proofs Without Words: Exercises In Visual Thinking, by Roger B. Nelsen, Mathematical Association of America,1993.

Adjust the size of the triangle by using the sliders at the top of the screen.

  • Call the length of the longer leg a, the length of the shorter leg b, and the length of the hypotenuse c.

Press the Arrange Four Copies button. This will show you an outline of how four copies of the triangle and a small square can be arranged to make a larger square. Press the Color the Copies button to color-code the pieces.

  • What is the area of this square, in terms of c?

Press the Rearrange the Shapes button to show how these five pieces can be arranged in a different configuration. Press the Behold! button to color-code the pieces.

  • Can you explain how this configuration is equal to a2 + b2?

If you need help answering the last question, press the Why It Works button. The thick blue segment divides the arrangement into two squares. Do you see that the area of one of the squares is a2 and the area of the other is b2?