Answer
$3, 4$ and $5$ miles.
Work Step by Step
Step 1. Assuming the length of the shorter leg is $x$ miles, we have the other leg as $x+1$
Step 2. Using the Pythagorean's Theorem, the hypotenuse is given by $\sqrt {x^2+(x+1)^2}=\sqrt {2x^2+2x+1}$
Step 3. Since the perimeter is 12 miles, we have $x+x+1+\sqrt {2x^2+2x+1}=12$ or $\sqrt {2x^2+2x+1}=11-2x$
Step 4. Taking the square on both sides of the above equation, we have $2x^2+2x+1=4x^2-44x+121$, $x^2-23x+60=0$ and $(x-3)(x-20)=0$
Step 5. Check the two solutions $x=3, 20$; we can see that only $x=3$ fits the problem. Thus the three sides are: $3, 4$ and $\sqrt {3^2+4^2}=5$ miles.
