Ajay starts from Point A and drives 6 km towards north. He then takes a left turn, drives 5 km, turns left and drives 15 km. He then takes a left turn and drives 12 km. He takes a final left turn, drives 9 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.)
Correct Answer :
7 km to the west
Solution :
To find the shortest distance and direction Ajay needs to drive to return to Point A from Point P, we can track his path step-by-step on a standard two-dimensional coordinate system. Let Point A be the origin .
Let's define the directions as follows:
- North corresponds to the positive y-axis direction.
- South corresponds to the negative y-axis direction.
- East corresponds to the positive x-axis direction.
- West corresponds to the negative x-axis direction.
Now, let's trace Ajay's movements sequentially:
1. Start at Point A:
Position is .
2. Drive 6 km towards North:
He moves 6 units up along the y-axis.
New position: .
3. Turn left and drive 5 km:
Facing North, a left turn points West (negative x-axis). He moves 5 units to the left.
New position: .
4. Turn left and drive 15 km:
Facing West, a left turn points South (negative y-axis). He moves 15 units down.
New position: .
5. Turn left and drive 12 km:
Facing South, a left turn points East (positive x-axis). He moves 12 units to the right.
New position: .
6. Take a final left turn and drive 9 km to Point P:
Facing East, a left turn points North (positive y-axis). He moves 9 units up.
Position of Point P: .
Now, we compare the final coordinates of Point P and the starting Point A:
- Point P is at .
- Point A is at .
To go from Point P back to Point A , Ajay must travel along the x-axis from to .
Distance = .
Since he is moving in the negative x-direction (from 7 to 0), the direction is West.
Therefore, Ajay should drive 7 km to the west to reach Point A again.
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