Monday, May 20, 2013

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 = m2n2, 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

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

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