Pythagorean Triple

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 : –

a² + b² = c²

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.

A² + B² = C²

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

According to the figure: –

AB² + BC² = AC²

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 : –

a² + b² = c²
3² + 4² = 5²
9 + 16 = 25

Some more examples to verify Pythagorean Triple : –

1. 5, 12 and 13

5² + 12² = 13²

25 + 144 = 169

169 = 169

Hence , it is a pythagorean triplet.

2. 7, 24 and 25

7² + 24² = 25²

49 + 576 = 625

625 = 625

Hence, it is a Pythagorean triplet.

Read More : Remember Multiplication Tables of Any Number Using Mental Mathematics

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 = x² – y²
• 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 = x² + y²

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Now, three formulas are generated from our assumptions:

a = x²– y²
b = 2xy
c = x²+ y²

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 = x²– y2, b = 2xy, c = x²+ y²

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 = x² – y²
a = 3² – 2²

9 – 4 = 5
a = 5
Computing b:
b = 2xy
b = 2 × 3 × 2
b = 12
Computing c:

c = x² + y²

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   a² + b² = c²  LHS:

5² + 12² = 25 +144 = 169

RHS : 132 = 169
Therefore, (5, 12, 13) are Pythagorean triples.