Let the starting point be at the origin with coordinates (0, 0).
The cyclist goes 12 km South:
Position: (0, -12)
The cyclist turns West and goes 16 km:
Position: (-16, -12)
The cyclist turns right (from West, right is North) and goes 15 km:
Position: (-16, -12 + 15) = (-16, 3)
The cyclist turns right (from North, right is East) and goes 12 km:
Position: (-16 + 12, 3) = (-4, 3)
The distance from the starting point (0, 0) to the final position (-4, 3) is:
d=(−4−0)2+(3−0)2
d=16+9
d=25
d=5 km
Therefore, the cyclist is 5 km from the starting point.