Understanding Direction and Distance Problems

This question asks us to find the final position of Lalit relative to his starting point after a series of movements in different directions. We need to determine both the straight-line distance and the direction from the start.

Step-by-Step Movement Analysis

Let's break down Lalit's journey:

1. Starts at a point (let's call it the origin).
2. Walks 9 km East.
3. Turns left. If facing East, a left turn means facing North. Walks 8 km North.
4. Again takes a left turn. If facing North, a left turn means facing West. Walks 3 km West.

Determining Net Displacement

We can analyze the movements along the East-West axis and the North-South axis separately.

• East-West Movement: Lalit moves 9 km East and then 3 km West. The net movement in the East-West direction is \( 9 \, \text{km (East)} - 3 \, \text{km (West)} = 6 \, \text{km East} \).
• North-South Movement: Lalit moves 8 km North. There is no movement South. The net movement in the North-South direction is \( 8 \, \text{km North} \).

So, from his starting point, Lalit's final position is 6 km to the East and 8 km to the North.

Calculating the Distance from Starting Point

The net movements form the two perpendicular sides of a right-angled triangle. The displacement (straight-line distance from the start) is the hypotenuse of this triangle. We can use the Pythagorean theorem to find this distance.

Let:

• \( \Delta x \) be the net displacement in the East-West direction = 6 km East
• \( \Delta y \) be the net displacement in the North-South direction = 8 km North
• \( D \) be the total distance from the starting point (the hypotenuse)

According to the Pythagorean theorem:

\( D^2 = (\Delta x)^2 + (\Delta y)^2 \)

Substituting the values:

\( D^2 = (6 \, \text{km})^2 + (8 \, \text{km})^2 \)

\( D^2 = 36 \, \text{km}^2 + 64 \, \text{km}^2 \)

\( D^2 = 100 \, \text{km}^2 \)

To find \( D \), we take the square root of both sides:

\( D = \sqrt{100 \, \text{km}^2} \)

\( D = 10 \, \text{km} \)

The distance from the starting point is 10 km.

Determining the Direction from Starting Point

Since the final position is 6 km East and 8 km North of the starting point, the direction is a combination of North and East. This direction is known as North East.

Final Answer Synthesis

Lalit is 10 km away from his starting point in the North East direction.

Comparing this with the given options, we find:

Revision Table: Key Concepts for Direction and Distance

Additional Information on Solving Direction and Distance Problems

Direction and distance questions often involve visualizing movements on a plane. Here are some tips:

• Draw a diagram: Sketching the path helps visualize the movements and the final position relative to the start.
• Use coordinates: Assign the starting point as (0,0). East is positive x, West is negative x, North is positive y, and South is negative y.
• Break down movements: Separate the total movement into net change along the horizontal (East-West) and vertical (North-South) axes.
• Apply Pythagorean theorem: If the net movements form a right angle, the displacement is the hypotenuse.
• Determine direction: Based on the signs of the net horizontal and vertical displacements, determine the final direction relative to the start (e.g., North East, South West).

Understanding turns is crucial:

• From North: Left is West, Right is East.
• From South: Left is East, Right is West.
• From East: Left is North, Right is South.
• From West: Left is South, Right is North.