##68##

The first thing to tackle here is how to express two consecutive even integers algebraically.

##2x## will give an even integer if ##x## is also an integer. The next even integer following ##2x## would be ##2x+2##. We can use these as the lengths of our legs but must remember that this will only hold true if ##x## is a (positive) integer.

Apply the :

##(2x)^2+(2x+2)^2=10^2##

##4x^2+4x^2+8x+4=100##

##8x^2+8x-96=0##

##x^2+x-12=0##

##(x+4)(x-3)=0##

##x=-43##

Thus ##x=3## since the side lengths of the triangle can’t be negative.

The legs are

##2xrArr6##

##2x+2rArr8##

##hypotenuserArr10##

A more intuitive way to do this problem is to recognize that a ##6810## triangle is just twice the size of the fundamental ##345## right triangle.