Step-by-Step Solution: Girl's Walk from School

This problem requires us to determine the final position of a girl relative to her starting point (the school gate) after a series of movements. We can track her position using a coordinate system, where the school gate is the origin $(0, 0)$. We'll define East as the positive x-direction, West as the negative x-direction, North as the positive y-direction, and South as the negative y-direction.

Tracking the Girl's Movements

Let's follow each step of her journey:

• Starting Point: School Gate at coordinates $(0, 0)$.
• 1. Walks 10 m East:
  • Movement: $+10$ m along the x-axis.
  • New Position: $(0 + 10, 0) = (10, 0)$.
• 2. Walks 4 m South:
  • Movement: $-4$ m along the y-axis.
  • New Position: $(10, 0 - 4) = (10, -4)$.
• 3. Walks 8 m West:
  • Movement: $-8$ m along the x-axis.
  • New Position: $(10 - 8, -4) = (2, -4)$.
• 4. Walks 2 m North:
  • Movement: $+2$ m along the y-axis.
  • New Position: $(2, -4 + 2) = (2, -2)$.
• 5. Turns right and walks 6 m:
  • At this point, she is facing North (from the previous step). Turning right means she now faces East.
  • Movement: $+6$ m along the x-axis.
  • New Position: $(2 + 6, -2) = (8, -2)$.
• 6. Turns left and walks 2 m:
  • She is currently facing East. Turning left means she now faces North.
  • Movement: $+2$ m along the y-axis.
  • New Position: $(8, -2 + 2) = (8, 0)$.
• 7. Turns left again and walks 6 m:
  • She is currently facing North. Turning left means she now faces West.
  • Movement: $-6$ m along the x-axis.
  • Final Position: $(8 - 6, 0) = (2, 0)$.

Calculating Net Displacement

Alternatively, we can sum the displacements along each axis:

• East-West (x-axis) Displacement:
  • Movements: $+10$ m (East), $-8$ m (West), $+6$ m (East), $-6$ m (West).
  • Net x-displacement = $10 - 8 + 6 - 6 = 2$ m.
• North-South (y-axis) Displacement:
  • Movements: $-4$ m (South), $+2$ m (North), $+2$ m (North).
  • Net y-displacement = $-4 + 2 + 2 = 0$ m.

The net displacement is $2$ m in the East direction and $0$ m in the North-South direction. The final coordinates are $(2, 0)$.

Final Position Relative to School Gate

The final position $(2, 0)$ means the girl is 2 meters from the origin along the positive x-axis. Since the positive x-axis represents East, she is 2 meters East of the school gate.

Therefore, the girl is located 2 m East from the school gate.