SOLUTION: ONE NUMBER IN A PYTHAGOREAN TRIPLE IS 25, FIND THE OTHER TWO

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Question 944729: ONE NUMBER IN A PYTHAGOREAN TRIPLE IS 25, FIND THE OTHER TWO
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is more than one answer to that.

A very popular Pythagorean triple (and the only one I remember) is %22%283%2C4%2C5%29%22 .
A (right) triangle with those measurements can be scaled up to another triangle whose measurements are other Pythagorean triples.
with a scale factor of 5 , we get highlight%28%22%2815%2C20%2C25%29%22%29 .

A Pythagorean triple like %22%2815%2C20%2C25%29%22 is considered a "multiple", while %22%283%2C4%2C5%29%22 is considered a "primitive".

A formula to generate the primitive Pythagorean triples is
2mn%22%2C%22m%5E2-n%5E2%22%2C%22m%5E2%2Bn%5E2 with positive integers m and n such that m%3En .

We can use that formula to generate another primitive Pythagorean triple that includes 5 .
m%5E2%2Bn%5E2=5<--->system%28m=2%2Cn=1%29 generates ,
which gives us the very popular %22%283%2C4%2C5%29%22 .
We cannot make 2mn=5 ,
but we can also make m%5E2-n%5E2=5<-->%28m%2Bn%29%2A%28m-n%29=5 with system%28m%2Bn=5%2Cm-n=1%29 <---> system%28m=3%2Cn=2%29 .
system%28m=3%2Cn=2%29 gives us .
Then, primitive Pythagorean triple %22%285%2C12%2C13%29%22 can be scaled up by a factor of 5 to highlight%28%22%2825%2C60%2C65%29%22%29 .

We can also use the formula to generate another primitive Pythagorean triple that includes 25 .
We cannot make 2mn=25 or m%5E2-n=%28m%2Bn%29%28m-n%29=25 .
We can only make m%5E2%2Bn%5E2=25 and that requires system%28m=4%2Cn=3%29 .
m%5E2%2Bn%5E2=25<--->system%28m=4%2Cn=3%29 generates .
That gives us highlight%28%22%287%2C24%2C25%29%22%29 .