A set of three positive integers that satisfy the Pythagorean theorem is a Pythagorean triple. 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. Here we will discuss Pythagorean triples formula which is a set of positive integers that satisfy the Pythagorean theorem.

**Pythagorean Triples Formula**

**Primitive and Non-Primitive Pythagorean Triples**

Pythagorean triples formula consists of three integers following the rules defined by the Pythagoras theorem. These triples are usually known as Pythagorean triples and are commonly written in the form of (a,b,c). And the triangle formed with these triples is called a Pythagorean triangle.

Pythagoras theorem is the square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the two sides The Pythagoras Theorem is expressed as, c^{2} = a^{2} + b^{2}

**Primitive Pythagorean Triples**

A set of numbers is considered as a primitive Pythagorean triple if all the three numbers in the triples have no common divisor other than 1. Primitive Pythagorean triples will always have one even number.** **

**Non – Primitive Pythagorean Triples**

A set of numbers is considered as a non- primitive Pythagorean triple if all the three numbers in the triples have a common divisor.

**The formula for Pythagorean Triples**

If a, b are two sides of the triangle and c is the hypotenuse, then, a, b, and c can be found out using the following formula-

a = (m^{2}-1), b = (2m), and c = (m^{2}+1)

**Properties**

A Pythagorean Triple always consists of all even numbers, or two odd numbers and an even number. If any constant number multiplies all the numbers in a triplet, then the resulting number makes the Pythagorean triples.

A Pythagorean Triple can never be made up of all odd numbers or two even numbers and one odd number because according to the properties of a square number, the square of an odd number is an odd number and the square of an even number is an even number.

The sum of two even numbers is an even number and the sum of an odd number and an even number is always an odd number. So, when the values of both a and b are even, so is c also even. Similarly when the values of either a or b are odd and the other is even, so the value of c will be odd.

### Pythagorean Triples Proof

Proof of Pythagoras theorem:

Area of square = (a+b)^{2}

Area of Triangle = \(\frac{1}{2}(ab)\)

Area of the inner square = b^{2}

The area of the entire square = \(4(\frac{1}{2}(ab)) \)+ c^{2}

Now we can conclude that

(a + b)^{2} = 4\((\frac{1}{2}(ab))\) + c^{2}.

or

a^{2} + 2ab + b^{2} = 2ab + b^{2}.

Simplifying, we get Pythagorean triples formula,

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

**Solved Examples**

Q.1: Check if (16, 63, 65) are Pythagorean triples?

Solution: Given, (16, 63, 65)

a = 16, b = 63, c = 65

Pythagorean triples formula used as,

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

LHS: c^{2 }= 65^{2} = 4225

RHS: a^{2} + b^{2} = 16^{2} + 63^{2} = 256 + 3969 = 4225

LHS = RHS

So, (16, 63, 65) is a Pythagorean triples.

Q.2. Write a Pythagorean triplet whose one member is 18

Solution: Let 2m = 18 , m = 9

Now, m^{2} – 1 = 9^{2} – 1 =81 – 1 = 80

And, m^{2} + 1 = 9^{2} + 1 = 81 + 1 = 82

So, the Pythagorean triple is 18, 80, 82

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