Question

Find the pythagorean triple for 9.

Answer 1

It looks like you need 2 numbers to form a pythagorean triple using Euclid's formula:

\[ a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2 \]

Answer 2

To be honest I never heard of it so I Googled it and that's what Wikipedia says.
"Euclid's formula[1] is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m and n with m > n. "

Answer 3

There is no pythagorean triple for 9..

Answer 4

If 9 is one of the legs of a right triangle (not the hypotenuse), then:\[9^2+b^2=c^2\Longrightarrow 81=c^2-b^2\Longrightarrow 81=(c-b)(c+b).\]Thus c-b and c+b must be divisors of 81. Since 81=3^4, the only cases we must consider are:\[c-b=1,c+b=81\]and \[c-b=3,c+b=27.\]The first case yields:\[c=41,b=40\]while the second yields:\[c=15,b=12.\]Hence the only pythagorean triples with 9 as a leg are:\[9^2+40^2=41^2\]and:\[9^2+12^2=15^2\]