Understanding Distance and Direction Problems

This type of question involves tracing a path based on given distances and turns, and then calculating the direct distance from the start point and the final facing direction.

Step-by-Step Path Analysis

Let's break down Arun's movement step by step. We can assume his home is the starting point, which we can place at the origin (0,0) of a coordinate system. North is positive Y, South is negative Y, East is positive X, and West is negative X.

1. Arun went 6 m from his home towards south: Starting from (0,0), he moves 6 m south. His position is now (0, -6). He is facing South.
2. ...turned left to cover 8 m: From (0, -6) and facing South, a left turn means facing East. He covers 8 m east. His new position is (0 + 8, -6) = (8, -6). He is now facing East.
3. Again he turned left and covered 6 m: From (8, -6) and facing East, a left turn means facing North. He covers 6 m north. His new position is (8, -6 + 6) = (8, 0). He is now facing North.
4. He again turned right and covered 4 m: From (8, 0) and facing North, a right turn means facing East. He covers 4 m east. His final position is (8 + 4, 0) = (12, 0). He is now facing East.

Calculating the Direct Distance from Home

The starting point (Home) is (0,0). The final position is (12,0).

The direct distance is the shortest straight line distance between the starting point and the ending point.

We can use the distance formula, but since the y-coordinate hasn't changed from the adjusted reference point (8,0 to 12,0, effectively), or more simply, from (0,0) to (12,0), it's a horizontal line along the x-axis.

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

Here, $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (12, 0)$.

Distance = $\sqrt{(12 - 0)^2 + (0 - 0)^2} = \sqrt{12^2 + 0^2} = \sqrt{144} = 12$ m.

Alternatively, we can visualize the net displacement. He moved 6m South and then 6m North, effectively cancelling out vertical displacement relative to the starting horizontal line. He moved 8m East, then 4m East. The total eastward displacement is 8 m + 4 m = 12 m. Since the vertical displacement is zero relative to the initial East-West line through Home, the direct distance is simply the total eastward displacement, which is 12 m.

Determining the Final Direction Faced

Let's look at the last turn:

• Before the last turn, he was at (8,0) and facing North.
• He turned right. When facing North, a right turn makes you face East.

Therefore, the final direction he was facing is East.

Summarizing the Results

Based on our analysis:

• The direct distance from his home is 12 m.
• The direction he was facing is East.

Let's check the options provided:

The results match Option 1.

Revision Table: Distance and Direction Concepts

Additional Information: Relative Turns

Understanding relative turns (left/right) is key in distance and direction problems. The direction you face after a turn depends entirely on the direction you were facing before the turn. Think about the cardinal directions in a cycle:

North → East → South → West → North ...

A right turn moves you one step forward in this cycle. A left turn moves you one step backward.

• Facing North: Right → East; Left → West
• Facing East: Right → South; Left → North
• Facing South: Right → West; Left → East
• Facing West: Right → North; Left → South

In this problem, Arun's turns were: South (Left → East), East (Left → North), North (Right → East). This confirms his final facing direction.