Pythagorean Triple: Formula & Examples

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson you will learn about primitive and non-primitive Pythagorean triples - sets of positive integers that satisfy the Pythagorean theorem. See if you can identify Pythagorean triples.

A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem: a^2 + b^2 = c^2. Integers are numbers that are written without decimal points or fractions. They may also be known as whole numbers.

The Pythagorean theorem shows the relationship of the squares of the sides of any right triangle - a triangle with a 90-degree, or square, corner.

Usually a and b refer to the two short sides of the triangle, and c refers to the longest side, the hypotenuse, as shown in the picture.

The Pythagorean Theorem
The Pythagorean Theorem

If you know the lengths of any two sides of the right triangle, then you can find the length of the third side using the Pythagorean theorem. For example, if a = 2 and b = 4. Then, using the theorem, you get 2^2 + 4^2 = c^2. In other words, 20 = c^2. So c equals the square root of 20. There is not an integer that equals the square root of 20. In fact, the square root of 20 is irrational. The best you can do is simplify it as two times the square root of 5. Using a calculator, you can get an approximate answer that looks simpler, but it is only approximate.

Sometimes, however, you can plug in positive integer values for two of the sides and get out another positive integer. For example, if a = 3 and b = 4, then a^2 + b^2 = 25 and so c = 5.

It feels a bit like a miracle when that happens. It's like dropping all your change into the change machine at the local bank and it just happening to equal exactly $100.00. You almost wonder if the change machine is rigged.

Some Pythagorean triples are:

(3, 4, 5)

(5, 12, 13)

(8, 15, 17)

(7, 24, 25)

(20, 21, 29)

Primitive and Non-Primitive Triples

If you were to take the a Pythagorean triple, for example (3,4,5) and multiply all the numbers in the triple by the same positive integer, you would get another Pythagorean triple. For example, multiplying by two, you get (6, 8, 10). Those numbers also satisfy the Pythagorean theorem: 6^2+8^2 = 10^2.

You can actually prove using algebra that it will always work because all you did was multiply both sides of the equation by 2^2.

The point here is that you can create an infinite number of Pythagorean triples simply by multiplying each triple by integers. On the other hand, the triples list above (and a bunch others like them) are special because they are not the result of multiplying another Pythagorean triple by a positive integer. These triples are called primitive Pythagorean triples. They do not come from other Pythagorean triples.

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