Solving Rohan's Directional Walk Problem

This problem involves tracking a person's movement through different directions and calculating their final distance from the starting point. Let's break down Rohan's journey step by step.

Tracking Rohan's Path

We can visualize Rohan's path or use coordinates to find his final position relative to his start.

• Step 1: Northward Movement

  Rohan starts at a point (let's call it the origin) and walks 3 kms towards North. His position is now 3 kms North of the start.

• Step 2: First Left Turn (Westward)

  He turns to his left. Since he was facing North, his left is West. He walks 2 kms West. His position is now 3 kms North and 2 kms West of the start.

• Step 3: Second Left Turn (Southward)

  He again turns left. Now facing West, his left is South. He walks 3 kms South. This movement cancels out his initial 3 kms Northward movement. He is now on the same horizontal line as his starting point, but 2 kms West of it.

• Step 4: Third Left Turn (Eastward)

  He turns left again. Facing South, his left is East. He walks for 3 kms. Starting from 2 kms West of the origin, walking 3 kms East means he first covers the 2 kms back to the starting point's North-South line, and then continues another 1 km East.

Calculating Final Distance from Start

After the final step, Rohan is 1 km East of the North-South line passing through his starting point. Since his North-South displacement is zero (3 km North - 3 km South = 0 km), his final position is exactly 1 km East of his starting point.

Therefore, the distance from the starting point is simply this eastward displacement.

Final Distance = 1 km

Using Coordinates (Alternative Method)

Let the starting point be (0, 0).

• After walking 3 km North: Position is (0, 3).
• Turning left (West) and walking 2 km: Position is (-2, 3).
• Turning left (South) and walking 3 km: Position is (-2, 3 - 3) = (-2, 0).
• Turning left (East) and walking 3 km: Position is (-2 + 3, 0) = (1, 0).

The starting point is (0, 0) and the final point is (1, 0).

The distance using the distance formula:

Distance = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Distance = $\sqrt{(1 - 0)^2 + (0 - 0)^2}$

Distance = $\sqrt{1^2 + 0^2}$

Distance = $\sqrt{1}$

Distance = $1$ km.

Both methods confirm that Rohan is 1 km away from his starting point.