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)

 

 

 

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

(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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Light, Math, & Color: 2014 Edition

For several years now I’ve taught a three-week course on making artglass windows. After learning the basic technique, I ask each student to research a math-related topic and illustrate it with an artglass window. This year’s group did exceptionally good work! The projects ranged from perennial favorites like the Pythagorean Theorem and tessellations to some new topics –  tesseracts, Borromean Rings, and Johnson’s Circle Theorem.

Pictures of their projects are below:

Webber: Borromean Rings

Farrington: Pythagorean Theorem

Cobb: Fibonacci Numbers

Howard: The Four-Color Theorem

Lund: Congruent Angles Formed by Parallel Lines and a Transversal

Boyd: Co-Eutrignon Theorem

Johnson: Angle Bisector Theorem

Compton: Sierpinski Triangle

Horner: Circle Inscribed in a Triangle

McCarthy: Tessellation

Hulme: Parallel Planes

Draughn: Circle Inscribed in a Triangle

Lazenby: Sierpinski Triangle

Nutter: Tesseract

Harrison: Exterior Angles in a Polygon

Triplett: Midpoint of Hypotenuse Properties

Burkart: Exterior Angles of a Polygon

Bennett: The Golden Ratio

Jenkins: Properties of Parallel Lines and Transversals

Trabue: Johnson’s Circle Theorem

A Classic Proof of the Pythagorean Theorem

My geometry students all know the Pythagorean Theorem, but they often don’t know why it’s true. Here’s one of my favorite visual proofs of the venerable theorem (I put this together using Geometer’s Sketchpad).

The Pythagorean Theorem