Choosing x = 3 and y = 1 in the Pythagorean identity results in the triple (8, 6, 10). The presence of 2xy ensures at least one leg of the triangle has an even-numbered length.
Let's choose two numbers, x = 3 and y = 1, where x replaces x in the Pythagorean identity, and y replaces y.
I chose x = 3 and y = 1 based on the idea of creating a Pythagorean triple where x^2 - y^2 and 2xy are part of the identity. These values were selected to demonstrate the simplicity of the resulting triple.
To find a Pythagorean triple using x = 3 and y = 1, substitute these values into the Pythagorean identity:
(3^2 - 1^2)^2 + (2 * 3 * 1)^2 = (3^2 + 1^2)^2
Simplifying:
(9 - 1)^2 + (6)^2 = (9 + 1)^2
8^2 + 36 = 10^2
64 + 36 = 100
This results in the Pythagorean triple (8, 6, 10).
At least one leg of the triangle must have an even-numbered length because of the presence of 2xy in the Pythagorean identity. Since 2xy involves multiplying two numbers, one of which is always even (either x or y), the product will be even. Therefore, the triple will contain an even number, ensuring one leg of the triangle has an even length.