Question 129112
There are two possibilities:


First, the longer leg could be 37, then the hypotenuse is 38, and the shorter leg is the square root of {{{38^2-37^2}}}.  {{{38^2-37^2=1444-1369=75}}}.  Since 75 is not a perfect square, this is not the correct answer.


The other possibility is that 37 is the short leg, x is the long leg, and x + 1 is then the hypotenuse.  Using Pythagoras:


{{{(x+1)^2=x^2+37^2}}}
{{{x^2+2x+1=x^2+1369}}}
{{{2x+1=1369}}}
{{{2x=1368}}}
{{{x=684}}}


So the Pythagorean triple is 37, 684, 685.  I'll leave it as a exercise for you to do the arithmetic and prove it.