Understanding the Directional Movement Problem

This question involves analyzing the paths taken by two individuals, Hemant and Anurag, after leaving school. They first walk together to a bus stop, and then diverge to their respective homes. We need to determine the direction of Anurag's home relative to Hemant's home.

Step-by-Step Path Analysis

Let's break down the movements of Hemant and Anurag from the bus stop, as the school-to-bus-stop path is common and serves as a starting point for their separate journeys.

Hemant's Path from the Bus Stop:

• Starts at the Bus Stop.
• Turns left and walks for \(500 \text{ m}\). Assuming standard directions, 'left' from North is West. So, \(0.5 \text{ km}\) West.
• From there, he turns towards his left and walks for \(2 \text{ km}\). If he was heading West, a left turn means turning South. So, \(2 \text{ km}\) South.
• Hemant reaches his home.

So, Hemant's home is located \(0.5 \text{ km}\) West and \(2 \text{ km}\) South of the Bus Stop.

Anurag's Path from the Bus Stop:

• Starts at the Bus Stop.
• Turns right and walks for \(250 \text{ m}\). Assuming standard directions, 'right' from North is East. So, \(0.25 \text{ km}\) East.
• From there, he turns right and walks for \(1 \text{ km}\). If he was heading East, a right turn means turning South. So, \(1 \text{ km}\) South.
• Anurag reaches his home.

So, Anurag's home is located \(0.25 \text{ km}\) East and \(1 \text{ km}\) South of the Bus Stop.

Determining Final Positions Relative to Bus Stop

Let's summarize the final positions of their homes with respect to the Bus Stop:

Finding the Direction of Anurag's Home from Hemant's Home

To find the direction of Anurag's home from Hemant's home, imagine standing at Hemant's home and looking towards Anurag's home. We need to determine the relative displacement from Hemant's position to Anurag's position.

• Horizontal Displacement: Hemant is \(0.5 \text{ km}\) West of the Bus Stop. Anurag is \(0.25 \text{ km}\) East of the Bus Stop. To get from Hemant's West position to Anurag's East position, you need to move East. The total eastward distance is \(0.5 \text{ km} (\text{to reach Bus Stop line}) + 0.25 \text{ km} (\text{from Bus Stop line to Anurag}) = 0.75 \text{ km}\) East.
• Vertical Displacement: Hemant is \(2 \text{ km}\) South of the Bus Stop. Anurag is \(1 \text{ km}\) South of the Bus Stop. To get from Hemant's position (\(2 \text{ km}\) South) to Anurag's position (\(1 \text{ km}\) South), you need to move North. The total northward distance is \(2 \text{ km} - 1 \text{ km} = 1 \text{ km}\) North.

From Hemant's home, Anurag's home is \(0.75 \text{ km}\) East and \(1 \text{ km}\) North.

A position that is both North and East relative to a starting point is in the North-East direction.

Conclusion on Direction

Based on the relative positions, Anurag's home is to the North-East of Hemant's home.

Revision Table: Analyzing Hemant and Anurag's Journeys

Additional Information on Directional Reasoning

Directional reasoning questions often require visualizing movements and calculating final positions based on sequences of turns and distances. It's helpful to:

• Establish a clear starting point and positive/negative directions (e.g., East is positive X, North is positive Y).
• Break down complex paths into simpler segments (e.g., North, South, East, West).
• Calculate the net displacement in each primary direction (North-South and East-West).
• Combine the net displacements to find the final position relative to the starting point.
• When finding the direction of point B from point A, calculate the displacement from A to B. If the displacement is, for instance, 'X units East' and 'Y units North', the direction is North-East.