Himank starts from Point A and drives 13 km towards West. He then takes a left turn, drives 3 km, turns left and drives 16 km. He then takes a left turn and drives 6 km. He takes a final left turn, drives 3 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are 90° turns only unless specified.)

Question

Prepp Team · Accepted Answer

Himank's Driving Directions Explained

This problem involves tracking a person's movement across a 2D plane and determining the final displacement required to return to the starting point. We need to find the shortest distance and the direction from Himank's stopping point (Point P) back to his starting point (Point A).

Movement Analysis

Let's break down Himank's journey step by step. We can use a coordinate system where Point A is the origin $(0, 0)$. We'll assume North is the positive y-direction, South is the negative y-direction, East is the positive x-direction, and West is the negative x-direction.

1. Start at Point A: Coordinates $(0, 0)$.

2. Drive 13 km towards West: Himank moves along the negative x-axis.

New position: $(0 - 13, 0) = (-13, 0)$.

3. Takes a left turn (now facing South) and drives 3 km: Himank moves parallel to the negative y-axis.

New position: $(-13, 0 - 3) = (-13, -3)$.

4. Turns left (now facing East) and drives 16 km: Himank moves parallel to the positive x-axis.

New position: $(-13 + 16, -3) = (3, -3)$.

5. Takes a left turn (now facing North) and drives 6 km: Himank moves parallel to the positive y-axis.

New position: $(3, -3 + 6) = (3, 3)$.

6. Takes a final left turn (now facing West) and drives 3 km: Himank moves parallel to the negative x-axis.

Final position (Point P): $(3 - 3, 3) = (0, 3)$.

Calculating Final Distance and Direction

Himank stopped at Point P, which has coordinates $(0, 3)$. He wants to return to Point A, which is at $(0, 0)$.

Starting Point for return journey: Point P $(0, 3)$.
Destination: Point A $(0, 0)$.

To find the shortest distance and direction, we calculate the difference in coordinates:

Change in x-coordinate ($\Delta x$): $0 - 0 = 0$.
Change in y-coordinate ($\Delta y$): $0 - 3 = -3$.

The distance is calculated using the distance formula, which simplifies here because the change in x is zero:

Distance $= \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(0)^2 + (-3)^2} = \sqrt{0 + 9} = \sqrt{9} = 3$ km.

The direction is determined by the sign of the coordinate changes:

$\Delta x = 0$ means no movement East or West is needed.
$\Delta y = -3$ means a movement of 3 units in the negative y-direction is needed. The negative y-direction corresponds to South.