A and B start from the same point. A cycles 10 km South, then turns to her right and cycles 9 km. B cycles 2 km North, then turn West and cycles 15 km, then turn to her left and cycle 12 km. Where is B with respect to A now?

Options

A. 6 km, West
B. 6 km, East
C. 24 km, West
D. 24 km, East

Answer

To solve the problem step by step, we will track the movements of A and B on a coordinate plane.

### Step 1: Determine A's Position
1. A starts at the origin point (0, 0).
2. A cycles 10 km South:
- Moving South means decreasing the y-coordinate.
- New position of A: (0, -10).
3. A then turns to her right (which is to the West) and cycles 9 km:
- Moving West means decreasing the x-coordinate.
- New position of A: (-9, -10).

### Step 2: Determine B's Position
1. B also starts at the origin point (0, 0).
2. B cycles 2 km North:
- Moving North means increasing the y-coordinate.
- New position of B: (0, 2).
3. B then turns West and cycles 15 km:
- Moving West means decreasing the x-coordinate.
- New position of B: (-15, 2).
4. B then turns to her left (which is South) and cycles 12 km:
- Moving South means decreasing the y-coordinate.
- New position of B: (-15, -10).

### Step 3: Determine the Relative Position of B with respect to A
1. A's final position is (-9, -10).
2. B's final position is (-15, -10).
3. To find the position of B with respect to A, we calculate the difference in their coordinates:
- The x-coordinate difference: -15 - (-9) = -15 + 9 = -6.
- The y-coordinate difference: -10 - (-10) = -10 + 10 = 0.
4. This means B is 6 km to the West of A.

### Conclusion
B is located 6 km to the West of A.