Pythagorean Triples: Definition, Formula, & Examples

A Pythagorean triple consists of three positive integers a, b, and c, which satisfy the condition a² + b² = c². A Pythagorean triple is commonly written in the form (a, b, c). A popular triple is (3, 4, 5). If the sides of a triangle form a Pythagorean triple, the triangle is called a Pythagorean triangle and it is always a right triangle. Pythagorean Triple is an important concept in both algebra and geometry. If students have an idea of common Pythagorean triples, they can solve problems related to right triangles more efficiently. We have provided detailed information on Pythagorean triples in this article. Read on to find out about its definition, formula and solved examples.

Pythagorean Triples: Definition

A Pythagorean triple has three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written as (a, b, c). If (a, b, c) is a Pythagorean triple, then (ka, kb, kc) is also a triple for any positive integer k. A primitive Pythagorean triple is a triple in which a, b and c are coprime numbers. Consider the positive integers 3, 4, and 5. If we evaluate these numbers, we will get,

3² + 4² = 5²

9+16 = 25

25=25

Hence, 3, 4, and 5 are the Pythagorean triple.

Formula of Pythagorean Triples

The formula of Pythagorean Triples is derived from the Pythagoras theorem. According to the Pythagoras theorem, In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If the longest side is c and the other two sides are a and b, then the formula of the triple is as follows

a² + b² = c²

If you want to find the triples, use the formula as given below:

a = m²-n²
b = 2mn
c = m²+n²

List of Common Pythagorean Triples

Check some of the commonly used Pythagorean triplets list tabulated below:

(3, 4, 5)	(5, 12, 13)	(8, 15, 17)	(7, 24, 25)
(20, 21, 29)	(12, 35, 37)	(9, 40, 41)	(28, 45, 53)
(11, 60, 61)	(16, 63, 65)	(33, 56, 65)	(48, 55, 73)
(13, 84, 85)	(36, 77, 85)	(39, 80, 89)	(65, 72, 97)

Pythagorean Triples: Examples

Some of the examples for Pythagorean triples are mentioned below:

Example 1: Find if (5,12,13) is a Pythagorean triple

Solution: Given, a= 5, b= 12 and c=13

The Pythagorean formula is a² + b² = c²

5² + 12² = 13²

25 +144=169

169=169

Therefore, (5,12,13) is a Pythagorean triple

Examples 2: Find if (8,15,17) is a Pythagorean triple

Solution: Given, a= 8, b= 15 and c=17

The Pythagorean formula is a² + b² = c²

8² + 15² = 17²

64 +225=289

289=289

Therefore, (8,15,17) is a Pythagorean triple

Also check,

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FAQs on Pythagorean Triples

Q. What is Pythagorean Triples?
A. Pythagorean triples are positive integers that satisfy the condition a²+b² = c². Here, a and b are the two sides of the right triangle and c is the hypotenuse.

Q. What is the formula used to find Pythagorean triples?
A. To find the triples, you can use the following formula given below:
a = m²-n²
b = 2mn
c = m²+n²

Q. What is the formula of Pythagorean triples?
A. The formula of Pythagorean triples is a²+b² = c².

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