Understanding Distance and Direction Problems

Distance and direction problems test your ability to follow a series of movements and determine the final position relative to the starting point. It's often helpful to visualize the path or draw a simple diagram.

Step-by-Step Movement Analysis

Let's track Rahul's journey step by step from his original position.

1. Starts at the origin: Assume Rahul starts at point O (Original Position).
2. Moves East 80 m: Rahul walks 80 m towards the east. Let's say he reaches point A. His position is now 80 m east of O.
3. Turns Right, Walks 30 m: From point A (East), turning right means turning towards the south. He walks 30 m south. Let's say he reaches point B. His position is now 80 m east and 30 m south of O.
4. Turns Right, Walks 30 m: From point B (facing South), turning right means turning towards the west. He walks 30 m west. Let's say he reaches point C. He moved 30 m west from A, which was 80 m east of O. So, his new east position relative to O is \(80 - 30 = 50\) m East. He is still 30 m south of O. His position is now 50 m east and 30 m south of O.
5. Turns North, Covers 30 m: From point C (facing West), turning north means moving upwards in our visualization. He walks 30 m north. He was 30 m south of O. Moving 30 m north cancels out the southward movement. Let's say he reaches point D (Final Position). His position is now 50 m east and \(30 - 30 = 0\) m south/north of O.

Calculating the Final Position

Let's represent the directions using coordinates, with the starting point at (0,0). East is positive x, North is positive y.

• Starting Point: (0, 0)
• Move East 80 m: (0 + 80, 0) = (80, 0)
• Turn Right (South), Walk 30 m: (80, 0 - 30) = (80, -30)
• Turn Right (West), Walk 30 m: (80 - 30, -30) = (50, -30)
• Turn North, Cover 30 m: (50, -30 + 30) = (50, 0)

Rahul's final position is at coordinates (50, 0).

Calculating Distance from Original Position

The original position was (0, 0). The final position is (50, 0).

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (50, 0)\).

Distance = \(\sqrt{(50 - 0)^2 + (0 - 0)^2}\)

Distance = \(\sqrt{(50)^2 + (0)^2}\)

Distance = \(\sqrt{2500 + 0}\)

Distance = \(\sqrt{2500}\)

Distance = \(50\)

So, Rahul is 50 meters away from his original position.

Summary of Movements

The movements can be summarized in a table:

The final position is (50, 0) which is 50 m east of the original position.

Revision Table: Key Concepts

Additional Information: Direction Concepts

Understanding how turns affect direction is crucial in distance and direction problems.

• If you are facing North: Right is East, Left is West.
• If you are facing East: Right is South, Left is North.
• If you are facing South: Right is West, Left is East.
• If you are facing West: Right is North, Left is South.

In this problem:

• Start (East): Turn Right → South
• Facing South: Turn Right → West
• Finally turns towards North: This means the next movement is in the North direction regardless of the previous facing direction (West).

Visualizing or sketching the path helps avoid confusion with turns.