drives 7 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He
takes a final right turn, drives 5 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
$90^{\circ}$ turns only unless specified.)
Understanding Babar's Driving Journey
This problem involves tracking a person's movement based on directions and distances to determine the final position relative to the starting point. We need to calculate the shortest distance and the required direction to return to the origin.
Mapping Babar's Movements Step-by-Step
Let's track Babar's position using a coordinate system, assuming Point A is the origin (0, 0). We'll consider West as the negative x-direction, East as the positive x-direction, North as the positive y-direction, and South as the negative y-direction.
- Start at Point A: Coordinates (0, 0).
-
Drives 14 km towards West: Babar moves along the negative x-axis.
- New position: ($0 - 14$, 0) = (-14, 0).
-
Takes a left turn, drives 7 km: Facing West, a left turn means he now faces South (negative y-direction).
- New position: (-14, $0 - 7$) = (-14, -7).
-
Turns right and drives 9 km: Facing South, a right turn means he now faces West (negative x-direction).
- New position: ($-14 - 9$, -7) = (-23, -7).
-
Takes a right turn and drives 7 km: Facing West, a right turn means he now faces North (positive y-direction).
- New position: (-23, $-7 + 7$) = (-23, 0).
-
Takes a final right turn, drives 5 km: Facing North, a right turn means he now faces East (positive x-direction).
- New position: ($-23 + 5$, 0) = (-18, 0). This is Point P.
Calculating the Final Distance and Direction
Babar's starting point, A, is at coordinates (0, 0). His final stopping point, P, is at coordinates (-18, 0).
To find the shortest distance between P and A, we calculate the difference in their coordinates.
- Change in x-coordinate: $0 - (-18) = 18$ km.
- Change in y-coordinate: $0 - 0 = 0$ km.
The net displacement is 18 km along the x-axis. Since Point P has an x-coordinate of -18 and Point A has an x-coordinate of 0, Point P is located 18 km to the West of Point A.
Therefore, to drive from Point P back to Point A, Babar needs to travel 18 km in the opposite direction of West, which is East.
Summary Table
| Leg | Direction | Distance (km) | Change in X | Change in Y | Coordinates |
|---|---|---|---|---|---|
| Start | - | - | 0 | 0 | (0, 0) (Point A) |
| 1 | West | 14 | -14 | 0 | (-14, 0) |
| 2 | South | 7 | 0 | -7 | (-14, -7) |
| 3 | West | 9 | -9 | 0 | (-23, -7) |
| 4 | North | 7 | 0 | +7 | (-23, 0) |
| 5 | East | 5 | +5 | 0 | (-18, 0) (Point P) |
Important Questions from Direction and Distance
- Johny starts from Point A and drives 6 km towards the North. He then takes a right turn, drives 17 km, turns right and drives 13 km. He then takes a right turn and drives 8 km. He takes a final right turn, drives 7 km and stops at Point P. How far (shortest distance) and towards which direction should he drive to reach Point A again? (All turns are 90 degrees turns only unless specified.)
- Town Q is to the east of Town P. Town R is to the south of Town Q. Town S is to the east of Town R. Town T is to the south of Town S. Town U is to the east of Town T. What is the position of Town U with respect to Town Q?
- Binoy starts from Point Y and drives 16 km towards the north. He then takes a right turn and drives 43 km, then turns right and drives 38 km. He then takes a right turn and drives 19 km. He takes a final right turn, drives 22 km and stops at Point Z. How far (shortest distance) and in which direction should he drive in order to reach Point Y again? (All turns are 90-degree turns only unless specified.)
- 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.)
- Shyam starts from Point A and drives 12 km towards east. He then takes a right turn,
drives 7 km, turns right and drives 5 km. He then takes a left turn and drives 9 km. He
takes a final right turn, drives 7 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
$90^{\circ}$ turns only unless specified.)
