Pythagorean triplets — lesson. Mathematics CBSE, Class 8.

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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

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

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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5. Pythagorean triplets

Theory: