Pythagorean Triples

Now that you have become familiar with the Pythagorean theorem, you can use it to find and verify Pythagorean triples. A Pythagorean triple (or Pythagorean triplet) is a set of three positive integers (a, b, and c, where a and b are legs and c is the hypotenuse) that can be the lengths of the three sides of a right triangle. Some of the most common Pythagorean triples include:

$(3,4,5)$

$(6,8,10)$

$(5,12,13)$

$(8,15,17)$

The standard format for writing a Pythagorean triple is $(a,b,c)$ , where $a<b<c$ and c corresponds to the hypotenuse of the right triangle. Not all textbooks use this format though, so go with however they're presented to you in class.

There isn't a trick to determine whether three positive integers will qualify as Pythagorean triples at a glance, so you'll have to operate on a case-by-case basis. You might be able to memorize some of the more common sets, but it's impossible to know them all. Fortunately, you can use your understanding of the Pythagorean theorem to easily verify whether any three positive integers qualify as Pythagorean triples.

What are Pythagorean triples?

Pythagorean triples are a set of positive integers that fit the rule based on the Pythagorean theorem, and they can be checked by using the same equation:

$a^2+b^2=c^2$

One of the most common Pythagorean triples is 3, 4, and 5. We can verify this Pythagorean triple by plugging it into the equation:

$3^2+4^2=5^2$

$9+16=25$

One thing to note is that once you have one set of Pythagorean triples, you can multiply each number by a constant to find infinitely more sets. For example, you can multiply 3, 4, and 5 by 2 to come up with 6, 8, and 10. Are these a Pythagorean triple? Let's find out:

$6^2+8^2={10}^2$

$36+64=100$

Yep, the math checks out! We can express this relationship mathematically by stating that if $(a,b,c)$ are Pythagorean triples then $(ka,kb,kc)$ are also Pythagorean triples provided that k is a positive integer.

There are also two types of Pythagorean triples. Primitive Pythagorean triples refer to sets with no common divisor larger than 1, meaning that the numbers are as small as they can be to preserve their correlation. For instance, $(3,4,5)$ are primitive Pythagorean triples because no number divides into all three evenly. In contrast, $(6,8,10)$ are not primitive Pythagorean triples because all three can be divided by 2 evenly and bring us back to the primitive Pythagorean triples $(3,4,5)$ . This means that you will never get primitive Pythagorean triples using the $(ka,kb,kc)$ format above because the results will always share at least a common divisor k.

That said, there are an infinite number of primitive Pythagorean triplets despite the restrictions just as there are an infinite number of right triangles even though the requirements are extremely specific. The possibilities in mathematics are limitless!

Do all right triangles correspond to Pythagorean triples?

No. In fact, the vast majority of right triangles do not correspond to Pythagorean triples. While all right triangles satisfy the Pythagorean Equation, remember that the definition of Pythagorean triples specifically mentions positive integers. For instance, consider a triangle with both legs measuring one and a hypotenuse measuring 2:

$a^2+b^2=c^2$

$1^2+1^2={\sqrt{2}}^2$

The equation results in an accurate statement, so we're working with the sides of a right triangle. However, $\sqrt{2}$ is irrational and not a positive integer, so we have not met the conditions for Pythagorean triples. Therefore, you cannot use Pythagorean triplets to determine whether a triangle is a right triangle.

At the same time, you don't need any triangles to plug a set of three positive integers into the Pythagorean theorem and see if they might qualify as Pythagorean triples. While most of the applications of Pythagorean triples tie back to geometry in some way, playing around with numbers with no particular end goal in mind can also prove beneficial and even fun.

What can I do with Pythagorean triples?

The earliest known application of Pythagorean triples is by ancient Babylonian land surveyors who used them to define plots of land as easily as possible. Archaeologists have discovered tablets related to both the sale of farmland and lists of Pythagorean triples to help surveyors perform math (no calculators back then remember!). Since you can easily increase the value of Pythagorean triples without affecting their correlation to each other by multiplying all three numbers by a constant, it was an easy way for the Babylonians to scale things. Ancient India is also known to have utilized Pythagorean triples in construction projects centuries before Pythagoras was even born.

Interestingly, Pythagoras himself used Pythagorean triplets to study the cosmos in abstract terms while the ancient Babylonians and Indians had far more practical uses for them. This goes to show how versatile mathematical applications can be.

Today, the primary use of Pythagorean triples is for determining the length of each side of a right triangle: a skill useful in fields such as navigation, construction, and cryptography. Many mathematicians also study Pythagorean triplets to search for patterns and properties unique to specific sets such as primitive Pythagorean triples. Of course, you'll probably get a better score on your next math test if you understand them too.

Pythagorean triples practice questions

a. Are $(9,16,25)$ Pythagorean triples? Why or why not?

Yes, they satisfy the $a^2+b^2=c^2$ equation

b. Are $(11,14,17)$ Pythagorean triples? Why or why not?

No, they don't satisfy the Pythagorean Equation

c. Are $(1,1,\sqrt{2})$ Pythagorean triples? Why or why not?

No, c is not an integer

d. $(8,15,17)$ represent Pythagorean triples. List three other sets of Pythagorean triples that can be derived from this information.

$(16,30,34);(24,45,51);(32,60,68);$ etc.

e. Are $(36,77,85)$ primitive Pythagorean triples?

Yes

f. Are $(15,20,25)$ primitive Pythagorean triples?

No, they share 5 as a common divisor

g. Are $(11,60,64)$ primitive Pythagorean triples?

No, they aren't Pythagorean triples at all

h. In your own words, explain how the measurements of a right triangle could satisfy the Pythagorean Equation but not represent Pythagorean triplets.

Pythagorean triplets must be three positive integers and the Pythagorean equation can have radical (non-integer) solutions.

Topics related to the Pythagorean Triples

Prime and Composite Numbers

Perfect Numbers

Relatively Prime Numbers

Flashcards covering the Pythagorean Triples

Finite Mathematics Flashcards

Practice tests covering the Pythagorean Triples

Finite Mathematics Diagnostic Tests

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