The hypotenuse of a right triangle is 10 inches. The lengths of the two legs are given by 2 consecutive even integers. How do you find the lengths of the two legs?

##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.