##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.