A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written, and a well-known example is. If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime
Example: (3, 4, 5)
By evaluating we get:
[tex] {3}^{2} + {4}^{2} + {5}^{2} [/tex]
9+16 = 25
Hence, 3,4 and 5 are the Pythagorean triples.
You can say “triplets,” but “triples” are the favoured term. Let’s start this topic by an introduction of Pythagoras theorem.
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If a triangle has one angle which is a right-angle (i.e. 90o), there exists a relationship between the three sides of the triangle.
If the longest side (called the hypotenuse) is r and the other two sides (next to the right angle) is called p and q, then:
[tex] \sf \: {p}^{2} + {q}^{2} + {r}^{2} [/tex]
or,
The sum of the squares of the other two sides is the same as the square of the longest side.
- How to Form Pythagorean Triples?
How to Form Pythagorean Triples?
As we know, the number can be an odd number or an even number. Now, let us discuss how to create the Pythagorean triples.
Case 1: If the number is odd:
Let us assume the number be “x”.
If “x” is odd, then the Pythagorean triple = x, (x²/2) – 0.5, (x2²/2) + 0.5.
Consider an example (7, 24, 25). Now, let us discuss how to form this Pythagorean triple.
Hre, x = 7, which is an odd number.
(x2/2) – 0.5 = (49/2) – 0.5 = 24.5 – 0.5 = 24
(x2/2) + 0.5 = (49/2) + 0.5 = 24.5 + 0.5 = 25
Hence, the Pythagorean triple formed is (7, 24,
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