##6,8##

##6,8##

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=-4,3##

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

##2xrArr6##
##2x+2rArr8##
##”hypotenuse”rArr10##

A more intuitive way to do this problem is to recognize that a ##6,8,10## triangle is just twice the size of the fundamental ##3,4,5## right triangle.