To create a Pythagorean triple using the Pythagorean identity, we need to choose two numbers to replace x and y. Let's say we choose x = 3 and y = 2.
We chose 3 for x and 2 for y arbitrarily. These values could have been any positive integers, but for simplicity, we can start with small numbers.
Now, we substitute these values into the Pythagorean identity:
(3^2 - 2^2)^2 + (2 * 3 * 2)^2 = (3^2 + 2^2)^2
(9 - 4)^2 + (12)^2 = (9 + 4)^2
(5)^2 + (12)^2 = (13)^2
25 + 144 = 169
We can see that the equation holds true. Therefore, the numbers 3 and 2 form a Pythagorean triple, specifically the triple (5, 12, 13), with 5 as one of the legs, 12 as the other leg, and 13 as the hypotenuse.
To find a Pythagorean triple using these numbers (x = 3 and y = 2), we square each number and then apply the Pythagorean identity. The sum of the squares of the two smaller terms will equal the square of the largest term (x^2 + y^2 = z^2). In this case, we found (3^2 + 2^2 = 13^2), which gives us a Pythagorean triple (5, 12, 13).
In a Pythagorean triple, at least one leg of the triangle must have an even-numbered length because when substituting x and y into the Pythagorean identity, the term (2xy)^2 will always be a multiple of 4. This results in the sum of two perfect squares, one being odd and the other being even. Hence, regardless of the specific values chosen for x and y, one of the legs will always have an even length.
Use the Pythagorean identity, Latex: (x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2\textsf{,} to create a Pythagorean triple
choose two numbers and identify which is replacing x and which is replacing y
how did you know which number to ues for x and y
Explain how to find a pythagorean triple using those numbers
Explain why at least one leg of the triangle that pythagorean triple represents must have an even-numbered length.
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