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Triplet means a set of three numbers.
A triplet \((a, b, c)\) of three natural numbers is called a Pythagorean triplets if it satisfies .
Example:
1. Check \((3, 4, 5)\) is a Pythagorean triplet.
Solution:
Here \(a = 3\), \(b = 4\) and \(c = 5\).
L.H.S. \(=\) \(a^2 + b^2\)
\(= 3^2 + 4^2\)
\(= 9 + 16\)
\(= 25\)
R.H.S. \(=\) \(c^2\)
\(= 5^2 = 25\)
So, \(25\) \(=\) \(25\)
L.H.S. \(=\) R.H.S.
Therefore, \((3, 4, 5)\) is a Pythagorean triplet.
2. Check \((6, 8, 9)\) is a Pythagorean triplet or not.
Solution:
Here \(a = 6\), \(b = 8\) and \(c = 9\).
L.H.S. \(=\) \(a^2 + b^2\)
\(= 6^2 + 8^2\)
\(= 36 + 64\)
\(= 100\)
R.H.S. \(=\) \(c^2\)
\(=\) \(9^2 = 81\)
So, \(100\) \(\ne\) \(81\)
L.H.S. \(\ne\) R.H.S.
Therefore, \((6, 8, 9)\) is not a Pythagorean triplet.
General form of Pythagorean triplet
Let us consider any natural \(a > 1\).
The triplet \((2a, a^2 - 1, a^2 + 1)\) will form a Pythagorean triplet.
The general formula to find a Pythagorean triplet is \((2a)^2 + (a^2 - 1)^2 = (a^2 + 1)^2\), for any natural number \(a > 1\).
Example:
Find a Pythagorean triplet one of whose least number is \(10\).
General form of Pythagorean triplet is \((2a, a^2 - 1, a^2 + 1)\).
Least number \((2a) = 10\)
\(a = \frac{10}{2}\)
\(a = 5\)
\(a^2 - 1 = 5^2 - 1 = 24\)
\(a^2 + 1 = 5^2 + 1 = 26\)
The triplet is \((10, 24, 26)\).
To check the triplet satisfies the Pythagorean relation:
\((2a)^2 + (a^2 - 1)^2 = (a^2 + 1)^2\)
\(10^2 + 24^2 = 26^2\)
\(100 + 576 = 676\)
\(676 = 676\)
Therefore, \((10, 24, 26)\) is a Pythagorean triplet.
Important!
L.H.S. \(=\) Left Hand Side
R.H.S. \(=\) Right Hand Side
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