Theory:

Objective:
   
In this lesson, we are going to learn about the Pythagorean triplets.
 
Pythagorean triplets:
 
A triplet \(a, b, c\) of three natural numbers \(a\), \(b\), and \(c\) is called a Pythagorean triplets, if a2+b2=c2.
 
For example:
 
I) \((3, 4, 5)\)
     
32+42=52Thatis9+16=25=52    
 
II) \((15, 8, 17)\)
 
152+82=172Thatis225+64=289=172
 
To find a Pythagorean triplet:
 
We can write the Pythagorean triplet using this formula 2a,a21,a2+1.
 
Here 2a,a21,a2+1 is Pythagorean triplet for any natural number a1.
 
Note: If \(a\) and \(b\) are relatively prime natural numbers such that ab and exactly one of them is even and other is odd, and the three numbers contain no common factors.
 
For example: Find a Pythagorean triplet whose one number is 14.
 
We know that 2a,a21,a2+1 is a Pythagorean triplet.
 
Now equate the given number with \(2a\) to find the value of \(a\).
 
2a=14Thena=142=7
 
Now substitute the \(a\) in the known formula 2a,a21,a2+1, then we get,
 
i)2a=2×7=14ii)a21=(72)1=48iii)a21=(72)+1=50
 
Therefore the Pythagorean triplets are \((14, 48, 50)\).