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Let's analyze the problem.

We have three riders (A, B, and C) traveling along different paths on a grid, but we are concerned with their *displacements*, not the paths themselves. Displacement refers to the straight-line distance between two points (initial and final positions), independent of the path taken.

### Breakdown:
1. **Rider A's Path:**
   - Starts at the Coffee Shop and moves to the Garage.
   - Path: 4 km east + 3 km south.
   - Using the Pythagorean theorem, the displacement is:
     \[
     \text{Displacement of A} = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{km}.
     \]
   
2. **Rider B's Path:**
   - Starts at the Garage and moves to the Coffee Shop.
   - Path: 5 km along the diagonal.
   - Since Rider B travels diagonally, the displacement is directly along the line. Thus, the displacement is 5 km, just like Rider A.

3. **Rider C's Path:**
   - Starts at the Coffee Shop and moves to the Garage, like Rider A, but Rider C takes a different path.
   - Path: 3 km south + 4 km east (the same as Rider A but in reversed order).
   - By the same calculation as Rider A, Rider C's displacement is also 5 km.

### Now, let’s assess the students’ statements:

- **Ali's statement**:
  - Ali says that Riders A and B have the same displacement because their path lengths are the same, and Rider C has the smallest displacement.
  - This is incorrect. Although Riders A and B travel different paths, their displacements are the same (5 km). Also, Rider C’s displacement is not the smallest; it's the same as A and B’s.

- **Bob's statement**:
  - Bob agrees that Rider C has the smallest displacement but claims that Riders A and B have different displacements because they move in opposite directions.
  - Bob is partially correct. The directions don’t matter for displacement; only the straight-line distance matters. All three riders have the same displacement of 5 km, so Bob is incorrect about Rider C having the smallest displacement.

- **Carol's statement**:
  - Carol says that the displacements of all three riders are the same since they travel between the same two points.
  - Carol is correct. Displacement depends only on the initial and final positions. All three riders traveled between the Coffee Shop and the Garage, so their displacements are the same (5 km).

### Conclusion:
You should agree with **Carol**. All three riders have the same displacement of 5 km.

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Would you like further details on any part of this explanation? Here are 5 questions to explore this further:

1. What is the importance of direction in calculating displacement?
2. Can you explain why paths can be different, but displacements can still be the same?
3. How does the Pythagorean theorem apply to find the displacement in this case?
4. Could the riders' displacements have been different if their starting or ending points changed?
5. How does the concept of vector displacement differ from scalar distance?

**Tip**: Displacement is always measured as a straight line between two points, regardless of the path taken.

Which of these students do you agree with regarding the displacement of Riders A, B, and C?

Explore how displacement is calculated for Riders A, B, and C on a motorcycle trip. Understand vector analysis, the Pythagorean theorem, and why all three riders have the same displacement of 5 km despite different paths. Suitable for Grades 9-12.

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