Complete General Reasoning Course:-
Pythagorean Triplet:- A Pythagorean triplet is a set of three positive integers (a, b, c) that perfectly satisfy the Pythagorean theorem: a2 + b2 = c2 .
-> The triangles formed by these Pythagorean triplets are shown below.
eg.- (a, b, c) = (3, 4, 5)
3 2 + 42 = 52
- How to Find Pythagorean Triplets: Different Methods:-
- The Scaling Method:- A fundamental Pythagorean triplet can be used to generate infinitely many new Pythagorean triplets by multiplying each of its three numbers by the same positive integer.
-> Let X be any positive integer. Then another Pythagorean triplet can be obtained by multiplying each element of the original triplet by X, as shown below.
eg.-
- The Scaling Method:- A fundamental Pythagorean triplet can be used to generate infinitely many new Pythagorean triplets by multiplying each of its three numbers by the same positive integer.
-> Let X be any positive integer. Then another Pythagorean triplet can be obtained by multiplying each element of the original triplet by X, as shown below.
i.e.
(X*a)2 + (X*b)2 = (X*c)2. The set (X*a, X*b, X*c)
that satisfies the Pythagorean theorem.- Finding Pythagorean Triplets Using the General Formula (Given One Number):-
If one number of the triplet is known (let's call it ), the remaining two numbers can be found using the following method.
If the given number is Even:
Let the even number be m ( first number of a triplet ) .-> Divide the number by 2 to get m/2 .-> The second number is m2 - 1 .-> The third number is m2 + 1 .Example: For the number 16, m/2 = 8. The other two numbers are 82 - 1 = 63 and 82 + 1 = 65. The triplet is (16, 63, 65) .
If the given number is Odd:Let the odd number be m ( first number of a triplet ) .-> The second number is (m2 - 1)/2 .-> The third number is (m2 + 1)/2 .Example: For the number 3, m = 3 . The other two numbers are (32 - 1)/2 = 4 and (32 + 1)/2 = 5. The triplet is (3, 4, 5) .
- Using Euclid's Formula (Generating new triples):-
To generate a new primitive Pythagorean triplet, select two positive integers m and n, where m > n > 0, and they are not both odd.Then, - a = m2 - n2
- b = 2mn
- c = m2 + n2
Example: If you choose m = 3 and n = 2 :Then, a = 9 - 4 = 5 ; b = 2*3*2 = 12 andc = 9 + 4 = 13The triplet is (5, 12, 13).
Let the even number be m ( first number of a triplet ) .
-> Divide the number by 2 to get m/2 .
-> The second number is m2 - 1 .
-> The third number is m2 + 1 .
Example: For the number 16, m/2 = 8. The other two numbers are 82 - 1 = 63 and 82 + 1 = 65. The triplet is (16, 63, 65) .
If the given number is Odd:
Let the odd number be m ( first number of a triplet ) .
-> The second number is (m2 - 1)/2 .
-> The third number is (m2 + 1)/2 .
Example: For the number 3, m = 3 . The other two numbers are (32 - 1)/2 = 4 and (32 + 1)/2 = 5. The triplet is (3, 4, 5) .
- Using Euclid's Formula (Generating new triples):-
To generate a new primitive Pythagorean triplet, select two positive integers m and n, where m > n > 0, and they are not both odd.
Then,
- a = m2 - n2
- b = 2mn
- c = m2 + n2
Example: If you choose m = 3 and n = 2 :
Then, a = 9 - 4 = 5 ; b = 2*3*2 = 12 and
c = 9 + 4 = 13
The triplet is (5, 12, 13).
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