Algebra + Area = A Nice Proof

In my Geometry classes, it is time to work with right triangles. Rather than simply present my students with the Pythagorean Theorem, I decided to have them prove it without them knowing that’s what they were doing.

I based the following activity on a proof I found at Alexander Bogomolny’s very useful site, Cut-The-Knot.

A Proof

  1. What is the total area of the four triangles below? (They are all congruent)

 4 rt. triangles

 

 

 2. If the triangles are rearranged into the square pattern below,

Pythagorean Proof Square

(a) In terms of a and b, what is the area of the “hole” in the middle?

 

 

(b) What is the overall area of the square pattern?

 

 

3. How can you use the results of parts 1 and 2 to prove a famous theorem?

 

I like this proof, because it’s very visual as well as algebra-based. The most challenging part for my students was figuring out the area of the square “hole”. Many assumed the side of the square was 1/2 of a, which is not correct. The satisfaction and excitement they had as they saw the pieces of the proof come together were well worth the effort, though! They were all familiar with the Pythagorean Theorem, but this was the first time any of them had actually proved it.

Click here for a pdf of the worksheet and here for the solution.

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