Person "P" walked 1 km towards the South, then turned 90 degrees anticlockwise and walked 2 km. He then turned left and walked 1 km, turned right and walked 2 km, and finally turned 90 degrees anticlockwise and walked 3 km. How far is he from his starting point?

Introduction:
This direction–sense question describes a path involving several turns, including explicit 90 degree anticlockwise rotations. The aim is to find the straight-line distance from the starting point after all the movements are completed. By tracking the person's position through a coordinate method, we can determine the final displacement and thus the required distance.

Given Data / Assumptions:
• P walks 1 km towards the South.• Then he turns 90 degrees anticlockwise (ACW) and walks 2 km.• Then he turns left and walks 1 km.• Then he turns right and walks 2 km.• Finally he turns 90 degrees anticlockwise and walks 3 km.• Flat terrain and standard compass directions are assumed.

Concept / Approach:
We assign coordinates: North as positive y, South as negative y, East as positive x, West as negative x. We also track the current facing direction at each step. A 90 degree anticlockwise turn is equivalent to a left turn when facing North, but we must always interpret it relative to the current facing direction. By successively updating the facing direction and coordinates, we end with a final coordinate from which the distance to the origin is easily computed using the Pythagorean theorem.

Step-by-Step Solution:
Step 1: Start at (0, 0) facing South. Walking 1 km South moves P to (0, −1).Step 2: He then turns 90 degrees anticlockwise from South. On a compass, anticlockwise from South takes us to East.Step 3: Walking 2 km East from (0, −1) moves him to (2, −1).Step 4: He now turns left from East, which makes him face North. Walking 1 km North takes him to (2, 0).Step 5: Next he turns right from North, which sends him facing East again. Walking 2 km East from (2, 0) moves him to (4, 0).Step 6: Finally, he turns 90 degrees anticlockwise from East, which points him North. Walking 3 km North from (4, 0) takes him to (4, 3).Step 7: The starting point is (0, 0) and the final point is (4, 3), which is a classic 3–4–5 right triangle.

Verification / Alternative check:
The net east–west movement is 4 km East and no movement West, so the x-displacement is +4 km. The net north–south movement is 1 km North (from step 4) plus 3 km North (from step 6) minus 1 km South (from step 1), giving 3 km North. The straight-line distance from (0, 0) to (4, 3) is sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 km, confirming that the final distance is 5 km.

Why Other Options Are Wrong:
Distances such as 4 km or 3 km correspond to only one side of the right triangle, not the hypotenuse. A distance of 2 km or 0 km would imply either a much smaller displacement or that he returned exactly to the starting point, which is not supported by the net east and north movement we computed. Only 5 km correctly reflects both components of his travel.

Common Pitfalls:
Students sometimes interpret anticlockwise turns as always meaning a turn to the left without considering the current facing direction, which can be misleading when your starting direction is not North. Others confuse the order of turns. Writing down a small table with columns for current direction, action, new direction, and new coordinates is a systematic way to avoid such mistakes.

Final Answer:
After all the movements, P is 5 km away from his starting point.