Question

Is the following triplets Pythagorean triplet? Show step by step solution.
$\left( {18, 79, 82} \right)$

Answer 1

Hint:
Here, we are required to show whether the given three natural numbers form a Pythagorean triplet or not. We will substitute the given numbers in the formula of Pythagorean Triplet, keeping in mind that on one side we have to add the square of two numbers and on the other side, we will take the square of the largest number among the three. If LHS$ = $RHS, then it will be a Pythagorean Triplet.

Formula Used: ${a^2} + {b^2} = {c^2}$

Complete step by step solution:
A Pythagorean triplet consists of three integers $a, b, c$ where $c$ is the largest integer, such that:
${a^2} + {b^2} = {c^2}$………………………………(1)
If we are given three numbers and we have to prove that whether they form a Pythagorean triplet or not, we should solve for ${a^2} + {b^2} = {c^2}$ such that the largest of the given numbers is substituted as $c$. If LHS$ = $RHS, then a Pythagorean Triplet is formed.
According to the question,
We have three numbers: $\left( {18,79,82} \right)$
Clearly, 82 is the largest number among them.
Hence, let $a = 18$, $b = 79$ and $c = 82$
Now, substituting these values in (1),
$ \Rightarrow {a^2} + {b^2} = {c^2}$
$ \Rightarrow {\left( {18} \right)^2} + {\left( {79} \right)^2} = {\left( {82} \right)^2}$
Now, finding the square of these numbers, we get,
$ \Rightarrow 324 + 6241 = 6724$
$ \Rightarrow 6565 \ne 6724$
Clearly,
${\left( {18} \right)^2} + {\left( {79} \right)^2} \ne {\left( {82} \right)^2}$
Hence, LHS$ \ne $RHS

Therefore, these numbers do not form a Pythagorean triplet.

Note:
A Pythagorean Triplet always fits in the formula ${a^2} + {b^2} = {c^2}$.
The smallest possible Pythagorean Triplet is 3,4,5 as these numbers satisfy:
${3^2} + {4^2} = {5^2}$
Also, whenever we find a Pythagorean triplet, it means that when we will try drawing a triangle, then it will definitely form a right angled triangle as it will obviously satisfy the Pythagoras Theorem.
Hence, these properties of the Pythagorean Triplet play a major role in Geometry.