**Pythagorean Triple **(Triples are known as triplets but triples is the majorly used term) can be defined as a set of 3 positive integers (integer is a whole number, it can be positive, negative or zero) a, b and c that fits in the pythagorean formula, which is : –

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

In other words, we can say that **Pythagorean Triple **are any 3 positive integers that fit into the pythagorean theorem.

Pythagorean theorem states that , in any right angled triangle, the square of the hypotenuse is equal to the sum of the squares of base and height.

These 3 sides of a right angled triangle A, B and C form the pythagorean triples.

A^{2} + B^{2} = C^{2}

Here A is altitude or height , B is our base and C is hypotenuse.

According to the figure: –

AB^{2} + BC^{2} = AC^{2}

The smallest **Pythagorean Triple **exists is: – 3, 4 and 5

Let’s check this: –

Suppose a = 3, b = 4 and c = 5.

According to the formulae : –

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

3^{2} + 4^{2} = 5^{2}

9 + 16 = 25

Some more examples to verify **Pythagorean Triple **: –

1. 5, 12 and 13

5^{2} + 12^{2} = 13^{2}

25 + 144 = 169

169 = 169

Hence , it is a pythagorean triplet.

2. 7, 24 and 25

7^{2} + 24^{2} = 25^{2}

49 + 576 = 625

625 = 625

Hence, it is a Pythagorean triplet.

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**How Can We Generate Pythagorean Triples?**

According to the figure, assume that: –

- a, and b are the legs of a right-angled triangle, a is the base and b is the height or altitude of the triangle.
- c is the hypotenuse of the right-angled triangle
- X and y are any two positive integers where x > y
- x and y are coprime numbers and both numbers should not be odd.

**Now, we will use ‘x’ and ‘y’ to find the exact values of the sides.**

- The length of side ‘a’ can be calculated by taking the difference between the squares of ‘x’ and ‘y’ which is expressed as

a = x^{2}– y^{2} - The length of side ‘b’ can be calculated by doubling the product of ‘x’ and ‘y’ which is expressed as : –

b = 2xy - At last, the length of side ‘c’ can be computed by the sum of the squares of x and y, which can be expressed as

c = x^{2}+ y^{2}

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**Now, three formulas are generated from our assumptions:**

a = x^{2}– y^{2}

b = 2xy

c = x^{2}+ y^{2}

Note: – To generate some random Pythagorean triples, assume random natural numbers x and y, such that x > y and compute the triples (a,b,c) such that

a = x^{2}– y2, b = 2xy, c = x^{2}+ y^{2}

Let us generate Pythagorean triples to better understand the steps: –

Given two integers are 2 and 3, generate a **Pythagorean Triple **from them?

Assume x = 3, y = 2

Hence x > y, x = 3, and y = 2.

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Now, simply put the values into the formulas of a, b, and c, to get the sides of the right triangle.

Computing a:

a = x^{2} – y^{2}

a = 3^{2} – 2^{2}

9 – 4 = 5

a = 5

Computing b:

b = 2xy

b = 2 × 3 × 2

b = 12

Computing c:

c = x^{2} + y^{2}

c = 32 + 22

c = 13

Let us check if our values for a = 5, b = 12, and c = 13 satisfy the Pythagorean theorem, which is a^{2} + b^{2} = c^{2 }LHS:

5^{2} + 12^{2} = 25 +144 = 169

RHS : 132 = 169

Therefore, (5, 12, 13) are Pythagorean triples.