Vignesh starts from Point Y and drives 17 km towards south. He then takes a left turn, drives 25 km. He then turns right and drives 9 km. He takes a final right turn, drives 25 km and stops at Point Z. How far (shortest distance) and towards which direction should he drive in order to reach Point Y again? (All turns are 90° turns only unless specified.)

Question

Prepp Team · Accepted Answer

Route Analysis

Let's track Vignesh's movement step by step using a coordinate system. Assume the starting Point Y is at the origin (0, 0). We will use coordinates (East, North).

Start at Y: Position is (0, 0).
17 km South: Vignesh moves downwards along the y-axis. His new position is (0, -17).
Left turn (West), 25 km: From South, a left turn faces West (negative x-direction). His new position is ($0 - 25$, -17) = (-25, -17).
Right turn (North), 9 km: From West, a right turn faces North (positive y-direction). His new position is (-25, $-17 + 9$) = (-25, -8).
Right turn (East), 25 km: From North, a right turn faces East (positive x-direction). His new position is ($-25 + 25$, -8) = (0, -8). This is the final Point Z.

Final Position Z

Point Z is located at coordinates (0, -8) relative to the starting Point Y (0, 0).

Calculating Return Path to Y

To find the shortest distance and direction from Point Z back to Point Y, we calculate the displacement between Z(0, -8) and Y(0, 0).

Net Displacement Calculation:
- Change in East-West direction (X-axis): $ \Delta x = x_Y - x_Z = 0 - 0 = 0 $ km.
- Change in North-South direction (Y-axis): $ \Delta y = y_Y - y_Z = 0 - (-8) = +8 $ km.
Shortest Distance: The distance is calculated using the Pythagorean theorem: $ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} $ $ d = \sqrt{(0)^2 + (8)^2} = \sqrt{0 + 64} = \sqrt{64} = 8 $ km.
Direction Determination: The net displacement along the Y-axis is +8 km. A positive value in this coordinate system indicates movement towards the North.

Therefore, Vignesh needs to drive 8 km towards North to reach Point Y again from Point Z.