Monday, March 2, 2015

Informal Proofs of the Pythagorean Theorem (Math 8)

We've explored some right triangles in class and found that if you make squares off each side of a right triangle and find their areas, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the legs. (For the picture below, the blue area + the red area = the green area.) This relationship is one way of expressing the Pythagorean Theorem.


Author: Pythagorean.svg: en:User:Wapcaplet
Derivative work: CP QQY (talk)
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

Therefore, the Pythagorean Theorem says that if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then 

a2 + b2 = c2

There are LOTS of different ways to prove the Pythagorean Theorem holds true for right triangles. You will explore two of them online and sketch your results in your math notebook.

  1. Open Hotmath's Pythagorean Theorem activity. (If that link doesn't work, try clicking on the Pythagorean Theorem activity at's Geometry activities. Don't click on "Geometry" at the side.)
  2. By default, the tab at the top for Dissection is chosen. In geometry, dissection is about cutting up a shape and rearranging it to make another shape. The square sticking off the longer leg is cut into four quadrilaterals. See if you can drag these pieces and the square sticking off the shorter leg to make the square sticking off the hypotenuse. If you get stuck, or just for a review, hit Animation.
  3. Next, click on the Chinese tab at the top to explore another proof. Follow the directions; if you get stuck, or to review, hit Animation.
  4. In your notebook, for both of these methods, sketch the "before" and "after" pictures that show a2 + b2 = c2.

(If you have technical difficulties, have a parent write a brief, signed note in your notebook explaining, and I will excuse you from the assignment and add a note to that effect.)

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