Pythagorean triple using odd numbers

PYTHAGOREAN TRIPLE USING ODD NUMBERS

Pythagoras theorem

In a right-angled triangle, the square of the hypotenuse is equal to the square of the other two sides i.e

a² + b² = c²

Where a or b = adjacent or opposite

And c = hypotenuse

Pythagorean Triple

A Pythagorean triple is a set of three (3) numbers; a, b, and c that are integers such that a² + b² = c².

Odd numbers

Odd numbers are numbers that are not divisible by 2. Examples include 3, 5, 7, 9, 11, etc

Having known what Pythagorean triple and odd numbers are, let’s proceed to find out how to form Pythagorean triple using odd numbers

Step 1: Assuming a is an odd number.

Step 2: Square it. That will be equal to a²

Step 3: Subtract one (1) from the number.

a² – 1

Step 4: Divide the result by 2. That is

(a² – 1)/2

Step 5: Since The resulting expression {(a² – 1)/2} are the same, add one (1) to one of them

{(a² – 1)/2 + 1}

{(a² – 1) + 2} / 2

(a² – 1 + 2) / 2

(a² + 1) / 2

So, the three expressions are

a, (a² – 1)/2, and (a² + 1)/2

Their squares become

a², {(a² – 1)/2}², and {(a² + 1)/2}²

Thus, a² + {(a² – 1)/2}² = {(a² + 1)/2}²

Examples of odd numbers ranging from one (1) to fifty (50) are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49. We will use the first five numbers excluding one (1) to illustrate the formula

Using 1 as a number will result in having one of the sides equal to zero, which will result in a triple (1, 0, 1), but definitely not a triangle, as no sides in a triangle is equal to Zero. – Law of the null effect – Kenneth

I. Let a = 3

3² + {(3² – 1)/2}² = {(3² + 1)/2}²

3² + {(9 – 1)/2}² = {(9 + 1)/2}²

3² + (8/2)² = (10/2)²

3² + 4² = 5²

9 + 16 = 25

25 = 25

II. Let a = 5

5² + {(5² – 1)/2}² = {(5² + 1)/2}²

5² + {(25 – 1)/2}² = {(25 + 1)/2}²

5² + (24/2)² = (26/2)²

5² + 12² = 13²