Ravi left home and cycled 10 km towards South, then turned right and cycled 5…

Concept

In direction-and-distance reasoning, only the NET straight-line displacement between the start and end points matters, not the total distance actually cycled. Resolve every leg of the journey onto two perpendicular axes, North-South and East-West: movements along the same axis in opposite directions cancel exactly, while movements along the same axis in the same direction add directly. Whenever the net displacement ends up lying entirely along a single axis, the straight-line distance back to the start is simply the sum of the movements along that axis.

Step-by-step application

1. Ravi starts at home and cycles 10 km South, so he is now facing South.

2. He turns right. Facing South, a right turn points him West; he cycles 5 km West.

3. He turns right again. Facing West, a right turn points him North; he cycles 10 km North.

4. He turns left. Facing North, a left turn points him West; he cycles 10 km West.

5. Net North-South displacement: the 10 km South leg and the 10 km North leg are equal and opposite, so they cancel completely, leaving zero net vertical displacement.

6. Net East-West displacement: both remaining legs (5 km and 10 km) point West, so they add directly to give the total net horizontal displacement.

7. Since the net displacement is purely horizontal, the straight-line distance from Ravi's final position back to home equals this net horizontal displacement: 15 km.

Cross-check

Cross-check with coordinates: place home at the origin (0, 0), with North as +y and East as +x. After the South 10 km leg: (0, -10). After the West 5 km leg: (-5, -10). After the North 10 km leg: (-5, 0). After the West 10 km leg: (-15, 0). The straight-line distance from (-15, 0) back to the origin (0, 0) is 15 km, confirming the value found above. This also matches the labelled diagram method, where the required distance AE is built from the two westward legs.

Hence, the straight-line distance Ravi must cycle to reach home is 15 km.