Question

Which one of the following is not a Pythagorean triplets?
A. $7,24,25$
B. $15,112,113$
C. $10,24,26$
D. $14,12,13$

Answer 1

Hint: First of all, let us recall the statement of the Pythagoras theorem. The square of the hypotenuse is equal to the sum of the squares of other two sides. Now, we can easily find Pythagorean triples

Formula to be used:
Let the longest side (hypotenuse) be $c$ . Then, the other two sides will be $a$ and $b$ .
Using Pythagoras theorem, we have ${a^2} + {b^2} = {c^2}$
Where $a,b$ and $c$ are positive integers..

Complete step-by-step answer:
${a^2} + {b^2} = {c^2}$ are Pythagorean triplets. Those triplets can be denoted as $\left( {a,b,c} \right)$ .
Here we are given some triplets. Among them, we need to find the triplets which are not Pythagorean triplets.

A) The given triplets is $\left( {7,24,25} \right)$
Here $a = 7$ ,$b = 24$ ,$c = 25$
Using ${a^2} + {b^2} = {c^2}$, we have
${7^2} + {24^2} = {25^2}$
$49 + 576 = 625$
$625 = 625$
Here we got the same number on both sides.
So, $\left( {7,24,25} \right)$ is a Pythagorean triplet.

B) The given triplets is $\left( {15,112,113} \right)$
Here $a = 15$ ,$b = 112$ ,$c = 113$
Using ${a^2} + {b^2} = {c^2}$, We have
${15^2} + {112^2} = {113^2}$
$225 + 12544 = 12769$
$12769 = 12769$
Here we got the same number on both sides.
So, $\left( {15,112,113} \right)$ is a Pythagorean triplet.

C) The given triplets is $\left( {10,24,26} \right)$
Here $a = 10$ ,$b = 24$ ,$c = 26$
Using ${a^2} + {b^2} = {c^2}$, We have
${10^2} + {24^2} = {26^2}$
$100 + 576 = 671$
Here we got the same number on both sides.
So,$\left( {10,24,26} \right)$ is a Pythagorean triplet.

D) The given triplets is $\left( {14,12,13} \right)$
Here $a = 14$ ,$b = 12$ ,$c = 13$
Using ${a^2} + {b^2} = {c^2}$, We have
${12^2} + {13^2} = {14^2}$
$114 + 169 = 196$
$313 \ne 196$
Here we didn’t get the same number on both sides.
So $\left( {14,12,13} \right)$ is not a Pythagorean triplet.

So, the correct answer is “Option D”.

Note: It is to be noted that the largest integer among triplets should be the hypotenuse. That is if we are given $\left( {14,12,13} \right)$, the largest integer $14$ will be the hypotenuse. Also, the integer belonging to the hypotenuse is always odd and one of the two sides will always be an odd number.