How do i explain how a set of whole numbers makes a pythagorean triple. I cant figure out how to explain this. All i can come up with is you take an odd number then square it and find the two numbers that can make that number.
If you are just testing if 3 numbers form a Pythagorean triple that is easy.
Assume the largest is the hypotenuse, (it has to be!)
and test
(smallest)^ + (middle one)^2 = largest^2
e.g. is 5, 12, 13 one of those?
is 13^2= 5^2 + 12^2 ?
is 169 = 25 + 144 ? YES, so you got one!
if you want to form them:
pick any two numbers, one even , one odd and call them m and n , so that m > n
so 2mn, m^2 - n^2, and m^2 + n^2 will always form a Pythagorean triple
e.g. let m = 5, n = 4
2mn = 40
m^2 - n^2 = 25 - 16 = 9
m^2 + n^2 = 25+16 = 41
is 41^2 = 40^2 + 9^2 ?? , YES
Explaining how a set of whole numbers forms a Pythagorean triple involves understanding the relationship between the numbers and the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).
Here's how you can explain the process to find Pythagorean triples:
1. Start with any two whole numbers, let's call them 'm' and 'n'. It's important to note that 'm' should be greater than 'n'.
2. Apply the following formulas to find the three numbers that make up the Pythagorean triple:
- The first number is given by: a = m^2 - n^2
- The second number is given by: b = 2mn
- The third number is given by: c = m^2 + n^2
3. The three numbers (a, b, and c) obtained from the formulas above form a Pythagorean triple, satisfying the Pythagorean theorem.
For example, let's say we want to find a Pythagorean triple using 'm = 3' and 'n = 2':
a = (3^2) - (2^2) = 9 - 4 = 5
b = 2 * 3 * 2 = 12
c = (3^2) + (2^2) = 9 + 4 = 13
Hence, the numbers 5, 12, and 13 form a Pythagorean triple, as 5^2 + 12^2 = 13^2.
It's interesting to note that if you choose different values of 'm' and 'n', you will obtain different sets of Pythagorean triples. This provides an infinite number of Pythagorean triples.
So, to summarize, by using the formulas derived from the Pythagorean theorem, you can generate Pythagorean triples by choosing suitable values for 'm' and 'n'.