A man travel 4 miles north east that 12 miles north, how far is he from the starting point
assuming you meant:
A man travel 4 miles north-east, then 12 miles north ....
Make your sketch and you can see that you can use the cosine law:
Distance^2 = 4^2 + 12^2 - 2(4)(12)cos 135°
= 16+144 - 96(-√2/2) = 227.88225..
distance = appr 15.1 miles