Your work is correct, but you should know why. Otherwise, you will not know when you can use this method.
Old-fashioned (known for last 2500 years) . North and west are at right angles. Distance west is 150 km; distance north is 120 km. Therefore, by Euclidean geometry, distance
[MATH]\sqrt{150^2 + 120^2} \approx 192.[/MATH]
Solving problems by geometry is hard. So new-fashioned method (known for less than 400 years). We use a Cartesian coordinate system. We pick an arbitrary but hopefully convenient origin (you chose Naples, but you could just as conveniently have chosen Rome). We use the distance formula
[MATH]\sqrt{(150 - 0)^2 + (120 - 0)^2} \approx 192.[/MATH]
In this very simple problem, the old-fashioned method is less work. But for problems that are only slightly harder, the old-fashioned method (using Euclid) is much harder.
In either case, you should sketch a diagram to see what you are doing.