Question

Which of the following triplets are Pythagorean?
 \[\left( {3,4,5} \right),\left( {6,7,8} \right),\left( {10,24,26} \right),\left( {2,3,4} \right).\]

Answer 1

Hint: For three numbers to be Pythagorean triplets, they should be in the form of \[\left( {2m,{m^2} - 1,{m^2} + 1} \right)\] . In this question for the given triplets we will check whether the given triplets are in the form of \[\left( {2m,{m^2} - 1,{m^2} + 1} \right)\] or not one by one and if they satisfy the following form then they are the Pythagorean triplets.
Here from the triplets an even number will be selected which will be equated to \[2m\] from which the value of \[m\] will be calculated.

Complete step-by-step answer:
Given the triplets to be checked are
 \[\left( {3,4,5} \right),\left( {6,7,8} \right),\left( {10,24,26} \right),\left( {2,3,4} \right).\]
Let the smallest even number be \[2m\]
We will check each of the given set one by one
I.For \[\left( {3,4,5} \right)\]
Let \[2m = 4\]
Hence \[m = 2\]
Now we will check for condition
 \[
  {m^2} - 1 = {\left( 2 \right)^2} - 1 = 4 - 1 = 3 \\
   \Rightarrow {m^2} + 1 = {\left( 2 \right)^2} + 1 = 4 + 1 = 5 \;
 \]
Therefore we can say the triplet \[\left( {3,4,5} \right)\] is a Pythagorean triplet.

II.Now for \[\left( {6,7,8} \right)\]
Let \[2m = 6\]
Hence \[m = 3\]
Now we will check for condition
 \[
  {m^2} - 1 = {\left( 3 \right)^2} - 1 = 9 - 1 = 8 \\
   \Rightarrow {m^2} + 1 = {\left( 3 \right)^2} + 1 = 9 + 1 = 10 \;
 \]
Therefore the triplet \[\left( {6,7,8} \right)\] is not a Pythagorean triplet.

III.Now for \[\left( {10,24,26} \right)\]
Let \[2m = 10\]
Hence \[m = 5\]
Now we will check for condition
 \[
  {m^2} - 1 = {\left( 5 \right)^2} - 1 = 25 - 1 = 24 \\
   \Rightarrow {m^2} + 1 = {\left( 5 \right)^2} + 1 = 25 + 1 = 26 \;
 \]
Therefore the triplet \[\left( {10,25,26} \right)\] is a Pythagorean triplet.

IV.Also for \[\left( {2,3,4} \right)\]
Let \[2m = 2\]
Hence \[m = 1\]
Now we will check for condition
 \[
  {m^2} - 1 = {\left( 1 \right)^2} - 1 = 1 - 1 = 0 \\
  {m^2} + 1 = {\left( 1 \right)^2} + 1 = 1 + 1 = 2 \;
 \]
Therefore the given triplets \[\left( {10,25,26} \right)\] are the Pythagorean triplets.
Therefore from the given set of triplets \[\left( {3,4,5} \right)\] and \[\left( {10,25,26} \right)\] are the Pythagorean triplets.
So, the correct answer is “\[\left( {3,4,5} \right)\] and \[\left( {10,25,26} \right)\] ”.

Note: A Pythagorean triplets consists of integers \[a,b,and,c\] where \[{a^2} + {b^2} = {c^2}\] . These triplets are written in the form \[\left( {a,b,c} \right)\] .
This is another method to check whether the given triplets are Pythagorean or not just by using the formula \[{a^2} + {b^2} = {c^2}\] , where the sum of the square of each first two number is equal to the square of the third number.
To check \[\left( {3,4,5} \right) \to \left( {a,b,c} \right)\]
 \[
  {a^2} + {b^2} = {c^2} \\
   \Rightarrow {3^2} + {4^2} = {5^2} \\
   \Rightarrow 9 + 16 = 25 \\
   \Rightarrow 25 = 25 \;
 \]