Kavita's Journey Analysis: Distance and Direction

This problem requires us to track Kavita's movement step-by-step and determine the final location of a pole relative to her starting point. We can visualize this using a standard coordinate system where East is the positive x-direction, North is the positive y-direction, West is the negative x-direction, and South is the negative y-direction.

Mapping Kavita's Path

Let's place Kavita's starting point, Point A, at the origin $(0,0)$ on the coordinate plane.

• Movement 1: Eastward Drive Kavita drives $7$ km East from A $(0,0)$. Her new coordinates become $(0+7, 0) = (7,0)$.
• Movement 2: First Left Turn She turns $90^\circ$ left, now facing North. She drives $12$ km in this direction. Her coordinates change from $(7,0)$ to $(7, 0+12) = (7,12)$.
• Movement 3: Second Left Turn She turns $90^\circ$ left again, now facing West. She drives $10$ km. Her coordinates change from $(7,12)$ to $(7-10, 12) = (-3,12)$. This point is designated as Point P.
• Pole's Position The pole is located $12$ km South of Point P $(-3,12)$. Moving South decreases the y-coordinate. So, the pole's coordinates are $(-3, 12-12) = (-3,0)$.

Determining the Pole's Location from Point A

We need to find the distance and direction of the pole (at coordinates $(-3,0)$) from the starting point A (at coordinates $(0,0)$).

• Horizontal Displacement: The difference in the x-coordinates is $-3 - 0 = -3$. This indicates a movement of $3$ km in the negative x-direction, which is West.
• Vertical Displacement: The difference in the y-coordinates is $0 - 0 = 0$. This indicates no movement in the North or South direction relative to Point A.

Combining these displacements, the pole is located $3$ km West of Kavita's starting point A.