Understanding Radha's Direction and Distance Problem

This question asks us to track Radha's movement step-by-step to find her final position relative to her starting position. We need to calculate both the straight-line distance between the two points and the direction of the starting point as seen from the finishing point. These are classic direction and distance reasoning problems.

Step-by-Step Analysis of Radha's Walk

Let's break down Radha's journey segment by segment. We can imagine her starting point as the origin (0,0) on a coordinate plane, where North is along the positive y-axis and East is along the positive x-axis.

1. Starting Point: Radha begins at (0, 0).
2. First Movement: She walks 14 km towards North. Her position becomes (0, 0 + 14) = (0, 14).
3. Second Movement: She turns right from North (which is East) and walks 22 km. Her position becomes (0 + 22, 14) = (22, 14).
4. Third Movement: She turns right from East (which is South) and walks 24 km. Her position becomes (22, 14 - 24) = (22, -10).
5. Fourth Movement: She turns left from South (which is East) and walks 26 km. Her position becomes (22 + 26, -10) = (48, -10).
6. Rotation and Final Movement: She is facing East. She rotates 90 degrees anti-clockwise. From East, 90 degrees anti-clockwise is North. She walks 24 km towards North. Her final position becomes (48, -10 + 24) = (48, 14).

Calculating Final Position Relative to Start

Radha's starting point was (0, 0).

Radha's finishing point is (48, 14).

To find the position of the finishing point relative to the starting point, we look at the coordinates (48, 14). The finishing point is 48 km East and 14 km North of the starting point.

Calculating Distance and Direction: Starting Point from Finishing Point

The question asks for the distance and direction of the starting point from the finishing point. This is the reverse direction from calculating the finishing point from the start.

Starting Point (S) = (0, 0)

Finishing Point (F) = (48, 14)

We need to find the vector from F to S. This is given by S - F.

Vector = (0 - 48, 0 - 14) = (-48, -14).

The distance between the starting and finishing points is the magnitude of this vector.

Distance = $\sqrt{(-48)^2 + (-14)^2}$

Distance = $\sqrt{2304 + 196}$

Distance = $\sqrt{2500}$

Distance = 50 km.

The direction of the starting point from the finishing point is given by the direction of the vector (-48, -14). A negative x-component (-48) indicates West, and a negative y-component (-14) indicates South.

Therefore, the starting point is in the South-West direction from the finishing point.

Summary of Movements and Positions

Starting Point: (0, 0)

Finishing Point: (48, 14)

Vector from Finishing to Starting Point: (0-48, 0-14) = (-48, -14)

Distance: $\sqrt{(-48)^2 + (-14)^2} = \sqrt{2304+196} = \sqrt{2500} = 50$ km.

Direction of (-48, -14): South-West.

Final Answer Summary

Based on our step-by-step analysis and calculations, the starting point is 50 km away from the finishing point, in the South-West direction.

Revision Table: Direction and Distance Concepts

When solving direction and distance problems like Radha's walk, remember the following key points:

• Establish a coordinate system (e.g., North = +y, East = +x).
• Map each movement segment to a vector change in position.
• Turning right/left depends on the current direction faced.
• Rotations change the direction faced before the next movement.
• Calculate the final position coordinates (x, y).
• To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the distance formula: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• To find the direction of point A from point B, consider the vector from B to A.
• (-x, -y) corresponds to South-West.
• (-x, +y) corresponds to North-West.
• (+x, -y) corresponds to South-East.
• (+x, +y) corresponds to North-East.

Additional Information on Direction and Distance Reasoning

Direction and distance questions are common in competitive exams. They test your ability to visualize paths and apply basic geometry and coordinate system understanding. Always read carefully whether the question asks for the finishing point's position from the start or the starting point's position from the finish, as this determines the direction calculation. Pay close attention to turns (right/left) and rotations (clockwise/anti-clockwise) relative to the current facing direction. Practice with various scenarios involving different angles and sequence of turns to master this topic.