Shorter leg = a
Longer leg = b
Hypotenuse = c

Hypotenuse - shorter leg = 8, so



Hypotenuse - longer leg = 4, so



Since it is a right triangle, the Pythagorean theorem
holds, so



So we have the system of three equations in three unknowns:



Solve the first one for a, and the second one for b:

, 

Substitute in the third equation




Can you solve that for c?  If not post again asking how.
It comes out as a factorable quadratic equation. 

You get two solutions for c, , and 

We must discard the hypotenuse of 4, since it would be
impossible for c, the longest side to differ from the
longer leg by 8.

Therefore 

Substituting in  
 and 

 and 

or

, 

So the perimeter = , choice a).

Edwin