Directions: Read the following information carefully and answer the questions given below:

Anita starts cycling from point A in the west direction. After travelling 4m, she reaches point B. From point B, she takes a left turn and rides 3m to reach point C. From point C, she turns right and travels 6m to reach point D. From point D, she takes a right turn and travels 9m to reach point E.

Benny starts cycling from point P in the north direction. After travelling 6m, he reaches point Q. From point Q, he takes a left turn and rides 5m to reach point R. From point R, he takes a left turn and rides 10m to reach point S. Now, from point S, he takes a right turn and travels 3m to reach point E.

In which direction is Benny’s initial position with respect to Anita’s final position?

The correct answer is North-East.

To determine the direction of Benny's initial position with respect to Anita's final position, we can trace the paths of both Anita and Benny on a standard 2D coordinate system where East is along the positive x-axis (+x), West is along the negative x-axis (-x), North is along the positive y-axis (+y), and South is along the negative y-axis (-y).

Step 1: Trace Anita's movement starting from point A

Let point A be at the origin:

$A=(0,0)$

1. Anita rides West for 4m to reach point B:
$B=(0-4,0)=(-4,0)$

2. From B, she turns left (facing West, turning left means going South) and rides 3m to reach point C:
$C=(-4,0-3)=(-4,-3)$

3. From C, she turns right (facing South, turning right means going West) and rides 6m to reach point D:
$D=(-4-6,-3)=(-10,-3)$

4. From D, she turns right (facing West, turning right means going North) and rides 9m to reach point E:
$E=(-10,-3+9)=(-10,6)$

Thus, Anita's final position E is located at (-10, 6).

Step 2: Trace Benny's movement starting from point P

Let Benny's starting position P be at coordinates $(x_P,y_P)$.

1. Benny rides North for 6m to reach point Q:
$Q=(x_P,y_P+6)$

2. From Q, he turns left (facing North, turning left means going West) and rides 5m to reach point R:
$R=(x_P-5,y_P+6)$

3. From R, he turns left (facing West, turning left means going South) and rides 10m to reach point S:
$S=(x_P-5,y_P+6-10)=(x_P-5,y_P-4)$

4. From S, he turns right (facing South, turning right means going West) and travels 3m to reach point E:
$E=(x_P-5-3,y_P-4)=(x_P-8,y_P-4)$

Step 3: Calculate coordinates of Benny's initial position P

Since both paths end at point E, we equate the coordinates:

$x_P-8=-10\Rightarrow x_P=-2$

$y_P-4=6\Rightarrow y_P=10$

Therefore, point P is located at coordinates (-2, 10).

Step 4: Determine the relative direction

To find the direction of Benny's initial position P(-2, 10) relative to Anita's final position E(-10, 6):

$\text{x-displacement}=-2-(-10)=+8$

$\text{y-displacement}=10-6=+4$

A positive x-displacement means East, and a positive y-displacement means North. Thus, Benny's initial position P is in the North-East direction with respect to Anita's final position E.