A man went 10 km towards South. Then turned East and covered 10 km, and turned…

Concept: In a Direction and Distance problem, only the straight-line distance between the start and end point matters, not the path walked between them. If a person's net displacement is p km along one axis (say, North-South) and q km along the perpendicular axis (East-West), these two legs meet at a right angle, so the direct distance between the two points is given by the Pythagorean theorem: distance = √(p2 + q2).

Application: Take East as the positive x-direction and South as the positive y-direction, starting at the origin (0, 0):

1. Walks 10 km South, reaching position (0, 10), now facing South.

2. Turns East and walks 10 km, reaching position (10, 10), now facing East.

3. Turns to the right while facing East, so now facing South; walks 10 km, reaching position (10, 20).

4. Turns to the left while facing South, so now facing East; walks 10 km, reaching position (20, 20).

Net displacement from the start: 20 km along the East-West axis and 20 km along the North-South axis.

Cross-check: Adding the four 10 km moves as vectors -- two South moves (10 + 10 = 20 south) and two East moves (10 + 10 = 20 east) -- gives the same endpoint (20 east, 20 south) as the step-by-step trace above, confirming both net perpendicular legs are 20 km each.

Result: distance = √(202 + 202) = √800 = 20√2 ≈ 28.28 km.