Which set of numbers represent a Pythagorean triple
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In other words, a² + b² = c².
To find a Pythagorean triple, we can look for three numbers that, when squared and added together, give a result that is a perfect square. Here are a few examples:
- (3, 4, 5): In this triple, 3² + 4² = 9 + 16 = 25, which is equal to 5².
- (5, 12, 13): Here, 5² + 12² = 25 + 144 = 169, which is equal to 13².
- (8, 15, 17): In this case, 8² + 15² = 64 + 225 = 289, which is equal to 17².
These are just a few examples of Pythagorean triples. There are infinitely many Pythagorean triples, and they can be generated using various formulas and methods. One well-known formula to generate Pythagorean triples is Euler's formula:
a = m² - n²,
b = 2mn,
c = m² + n²,
where m and n are positive integers, and m > n. This formula guarantees that a, b, and c will form a Pythagorean triple. For example, choosing m=2 and n=1 in Euler's formula gives us (3, 4, 5), which is one of the examples mentioned earlier.