The Pythagorean Relationship

By Saler Axel, RME Research Assistant 

Math has a reputation of being dull. Luckily, there are some fun math holidays that exist throughout the year. Two popular ones are Pi Day (3/14) and Mole Day (10/23). Last week was 5-12-13 Triangle Day! What makes the 5-12-13 right triangle worth celebrating?

Let’s spend time considering special right triangles, which are some of geometry’s extraordinary shapes. A right triangle contains sides lengths that can be calculated using the Pythagorean Theory, a² + b²= c². We will spend time discussing right triangles like the 5-12-13 right triangle.

Side-based special triangles, such as a 5-12-13 right triangle, contain proportionate side lengths that make computing easier. Called Pythagorean Triples, these triangles contain angles with degrees that are never rational numbers. If students understand the relationships of a special right triangle’s side lengths, they can calculate other side lengths in geometric problems without having to employ difficult strategies.

An easy way to calculate Pythagorean Triples: a = m²– n², b = 2mn, c = m²+ n². where m and n are relatively prime positive integers and m>n.

Below are some things that you can do in your classroom to celebrate this extraordinary shape.

• Challenge students to calculate scaled examples of 5-12-13 triangles.

• Draw a 5-12-13 right triangle on grid paper. (An example of a 3-4-5 triangle is below.) Have your students make a square from each side. The diagram should have a 5•5 square on the left, a 12•12 square on the bottom, and a 13•13 square off of the hypotenuse. Encourage your students to measure the number of square units. They will discover that 5²+ 12²= 13². Then ask your students to try the same activity with an isosceles triangle (or any other type of triangle except a right triangle). This will help them understand that if they measure the squares, the sides will not make a right triangle.
• Here, the two squares together are a "proof without words." Here we see that:
  
  a² + 2ab+b²= c²+ 2ab

a²+ b²= c²

Other common Pythagorean Triples include those with side length ratios of: 3-4-5, 8-15-13, 7-24-25, and 9-40-41, though the possibilities are endless using the formula (3n)²+ (4n)²= (5n)². For an extensive list of Pythagorean Triples, visit www.mathisfun.com/numbers/pythagorean-triples.html.

How can you tailor these and other classroom lessons to expand your students’ thinking about special right triangles and their importance in geometric calculations?