Pythagorean Triplet


 
 
Concept Explanation
 

Pythagorean Triplet

Property :For any natural number m greater than 1 the Pythagorean triplet are represented as

(2m,m^{2}-1,m^{2}+1)

Proof: In order to prove that (2m,m^{2}-1,m^{2}+1) is a Pythagorean triplet, it is sufficient to prove that

                  (2m)^{2}+(m^{2}-1)^{2}=(m^{2}+1)^{2}

We have,   (2m)^{2}+(m^{2}-1)^{2}=4m^{2}+m^{4}-2m^{2}+1^{2}

                                                     =m^{4}+2m^{2}+1^{2}

                                                     =(m^{2}+1)^{2}

Hence, (2m,m^{2}-1,m^{2}+1) is Pythagorean triplet for any natural number m > 1.

Remark:  If m and n are relatively prime natural numbers such that m > n and exactly one of them is even and other is odd, then (2mn,m^{2}-n^{2},m^{2}+n^{2}) is a primitive Pythagorean triple.

Here, the word primitive means that the three numbers contain no common factor.

Illustration: Write a Pythagorean triplet whose one member is:

      (i) 6                       (ii) 18

Solution: For any natural number m > 1, we have

                     2m,m^{2}-1,m^{2}+1 as a Pythagorean triplet.

(i) Here,

       2m=6Rightarrow m=3.

therefore ;m^{2}-1=3^{2}-1=9-1=8  and m^{2}+1=3^{2}+1=9+1=10     

Hence the pythagorean triplet is (6,8,10)

(ii)  Here,

2m=18Rightarrow m=9.

therefore ;m^{2}-1=9^{2}-1=81-1=80     and m^{2}+1=9^{2}+1=81+1=82

Hence the pythagorean triplet is (18,80,82)

                                       

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Choose a Pythagorean triplet whose one member is 4.

Right Option : A
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Explanation
Question : 2

What is the pythagorean triplet if one member is 16

Right Option : B
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Explanation
Question : 3

Choose a Pythagorean triplet whose one member is  15

Right Option : B
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Explanation
 
 


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