A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle.

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Answer:

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• A Pythagorean triplet consists of three positive integers a, b, and c, such that a2 + b2 = c2.

• Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

• A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle

Formula

• To find the Pythagorean triples, the following formula is used. If a, b are two sides of the triangle and c is the hypotenuse, then, a, b, and c can be found out using this-

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• These values result in a right-angled triangle with sides a, b, c.

Notes:

• (i) m, n and k are any two positive integers

• (ii) m > n

• (iii) m and n are coprime and both should not be odd numbers

Solves Example

Question:

• Check if (7, 24, 25) is a Pythagorean triple.

Solution:

Given,

• Pythagorean triple = (7, 24, 25)

• a = 7, b = 24, c = 25

• The Pythagorean triples formula is, c2 = a2 + b2

• LHS: c2 = 252 = 625

• RHS: a2 + b2 = 72 + 242 = 49 + 576 = 625

• LHS = RHS

So, (7, 24, 25) is a Pythagorean triple

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