# Pythagorean Triples Formula

Pythagorean triples formula** **comprises three integers that follow the rules defined by the Pythagoras theorem. The group of these triples is referred to as the Pythagorean triples and is commonly written in the form of (a, b, c). The triangle formed with the sides having these triples as the dimensions is called a Pythagorean triangle. Let us understand the Pythagorean triples formula in detail using solved examples in the following section.

## What Is Pythagorean Triples Formula?

Pythagorean triples formula is used to find the triples or group of three terms that satisfy the Pythagoras theorem. We know that when a, b c are the base, perpendicular and the hypotenuse of a right-angled triangle, then by Pythagoras' theorem we have: c^{2} = a^{2}+b^{2}. The Pythagorean triples formula can be given as,

- a = m
^{2}- n^{2} - b = 2mn
- c = m
^{2}+ n^{2}

where,

- a, b = Base, and perpendicular of a right-angled triangle
- c = Hypotenuse of a right-angled triangle
- m, n are any two positive integers; m > n
- m and n are coprime and both should not be odd numbers

### Pythagorean Triples Formulas

Any one of the following formulas is used based on the condition and the requirement of finding the Pythagorean triples.

- If any number "m" of a Pythagorean triple is given, then the other two numbers can be evaluated using the trick by taking the triples as (2mn, m
^{2}- 1, m^{2}+ 1) - To generate random Pythagorean triples, consider random natural numbers m and n, such that m >n and determine the triples (a,b,c) in such a condition that a = m
^{2}- n^{2}, b = 2mn, c= m^{2}+ n^{2} - If a number (n) odd is given, then the Pythagorean triples are of the form, (n, (n
^{2}/2 - 0.5) and (n^{2}/2 + 0.5)). - If a number (n) even is given, then the Pythagorean triples are of the form = n, (n/2)
^{2}-1), ((n/2)^{2}+1).

### Pythagorean Triples Formula Verification

We can randomly generate Pythagorean triples using the Pythagorean triples formula. When a = m^{2}−n^{2}, b = 2mn, and c = m^{2}+n^{2}. When a = m^{2}- n^{2}, b = 2mn and c = m^{2}+n^{2},^{ }we can verify the Pythagorean triples formula using the Pythagorean theorem: c^{2} = a^{2 }+ b^{2 }

Consider LHS: c^{2} = (m^{2}+n^{2})^{2}

c^{2} = m^{4 }+^{ }n^{4 }+^{ }2m^{2}n^{2}

Consider RHS: a^{2}+b^{2 }= (m^{2}- n^{2} )^{2 }+ (2mn)^{2}

= m^{4 }+^{ }n^{4 }-^{ }2m^{2}n^{2 }+ 4m^{2}n^{2 }

a^{2}+b^{2 }= m^{4 }+^{ }n^{4 }+^{ }2m^{2}n^{2}

LHS= RHS

Thus c^{2} = a^{2 }+ b^{2}

Let us have a look at a few solved examples on the Pythagorean triples formula to understand the concept better.

## Examples on Pythagorean Triples Formula

**Example 1: **Check if (5, 12, 13) is a Pythagorean triple.

**Solution:**

To find: Check whether (5, 12, 13) is a Pythagorean triple.

Given: a = 5; b = 12; c = 13

(13 is the longest side and is considered to be the hypotenuse.)

Using the Pythagorean triples formula, we know that a Pythagorean triple satisfies Pythagoras' theorem: c^{2} = a^{2}+b^{2}

L. H. S. = c^{2} = 13^{2} = 169

R. H. S. = a^{2 }+ b^{2 }= 5^{2 }+ 12^{2} = 25 + 144 = 169

Since the given values satisfy the Pythagoras' theorem,(5, 12, 13) is a Pythagorean triple.

**Answer:** **(5, 12, 13) is a Pythagorean triple.**

**Example 2:** Evaluate and find why 40, 76, 86 is not a Pythagorean triple.

**Solution:**

Using Pythagorean Triples formula, a^{2} + b^{2} = c^{2}

= 40^{2} + 76^{2} = 1,600 + 5,776

= 7,376 ≠ 86^{2}

**Answer: 40, 76, 86 is not a Pythagorean triple.**

**Example 3:** If (x, 40, 41) is a Pythagorean triple, determine the value of x using the Pythagorean triple formula?

Solution:

Using the Pythagorean Triples formula:

a^{2} + b^{2} = c^{2}

Replacing a by x, b by 40, and c by 41 in the formula we have,

⇒ x^{2} + 40^{2} = 41^{2}

⇒ x^{2} + 1,600 = 1,681

⇒ x^{2} = 1,681 - 1,600 = 81

⇒ x = √81 = 9

## FAQ's on Pythagorean Triples Formula

### What Is The Pythagorean Triples Formula When one Side Is Given?

The Pythagorean triples formula is applied based on the requirement and the conditions. If one side of a right triangle is given, then the other two sides are determined on the basis of the given number odd or even.

- If a number (n) odd is given, then the Pythagorean triples are of the form, (n, (n
^{2}/2 - 0.5) and (n^{2}/2 + 0.5)). For example, consider 3. The triples are (3, (9/2 - 0.5), (9/2 + 0.5)). Finally, we get (3, 4 and 5). - If a number (n) even is given, then the Pythagorean triples are of the form, (n, (n/2)
^{2}-1), ((n/2)^{2}+1). For example, consider 8. The triples are (8, ((4)^{2}- 1), ((4)^{2}+ 1)). Finally, we get (8, 15, and 17)

### What Is Pythagorean Triples Formula Using random numbers?

Let us take two random numbers m = 4 and n = 3, where m> n. According to the Pythagorean triples formula, a = m^{2}- n^{2}, b = 2mn and c = m^{2}+n^{2}.

a = 4^{2}- 3^{2 }= 16 - 9 = 7,

b = 2(4)(3)= 24

and c = 4^{2}+3^{2 }= 16 +9 = 25

Thus the Pythagorean triples are (7, 24, 25) using random numbers.

### How Do you Find the Pythagorean Triplet of 10?

We find that 10 is even. Thus by the Pythagorean triples formula, we get (10, (10/2)^{2} + 1, (10/2)^{2} - 1) = (10, (5)^{2} - 1, (5)^{2} +1) = (10, 24, 26)

### What Are The Pythagorean Triples Whose Smallest Number is 8?

Let us consider 8 as the base side. Thus using the Pythagorean triples formula, we find 8 = 2m. This implies m = 4. The other two triples are m^{2} + 1 and m^{2} - 1.

Thus we evaluate them as 4^{2} + 1 and 4^{2} - 1 = 17 and 15.

Thus the triples are (8,15,17)