Question

Help in 30 minutes

Use the Pythagorean identity, (×^2 - y^2)^2 + (2xy)^2 = (×^2 + y^2)^2 to create a Pythagorean triple.

Follow these steps:

Choose two numbers and identify which is replacing a and which is replacing y.

How did you know which number to use for a and for y?

Explain how to find a Pythagorean triple using those numbers.

Explain why at least one leg of the triangle that the Pythagorean triple represents must have an even-numbered length.

Answer 1

Let's choose two numbers, 3 and 4, to replace × and y respectively.
We can see that 4 is replacing y because it is a perfect square, whereas 3 is not.

Now we can substitute these values into the Pythagorean identity:

(3^2 - 4^2)^2 + (2*3*4)^2 = (3^2 + 4^2)^2

(-7)^2 + (24)^2 = (9)^2 + (16)^2

49 + 576 = 81 + 256

625 = 625

Therefore, a Pythagorean triple has been created: (3, 4, 5).

We know that at least one leg of the triangle that the Pythagorean triple represents must have an even-numbered length because one of the Pythagorean triple's numbers must be divisible by 2. This is because if one of the legs is odd, then the other leg and the hypotenuse would be odd as well, and the sum of two odd numbers is always an even number. However, the square of an even number is always a multiple of 4. Therefore, at least one of the legs must be even in order to satisfy the Pythagorean identity.