https://file.aiquickdraw.com/mathbot/file/1726792225089wonadab5.jpg

The problem depicted is about finding the shortest walking distance between point A and point B in a grid-like structure, similar to Manhattan's city blocks. The two routes presented are:

1. **Yellow zig-zag path:** This involves traveling in a series of right angles, alternating between horizontal and vertical movements.
2. **Blue-green path (single-turn path):** This involves one horizontal movement (east) followed by one vertical movement (north).

### Key Considerations:

- In both paths, the total movement in the horizontal (east) direction and the vertical (north) direction remains the same, as both paths lead from A to B.
- The **zig-zag path** has more turns and intermediate changes in direction, but the overall distance traveled in each direction (north and east) is still identical to the straight-line path.

### Analysis:
- The blue-green path consists of one straight movement east and one straight movement north, without any unnecessary turns.
- The zig-zag path takes the same total number of steps in the horizontal and vertical directions, but because of the extra turns, it may appear longer. However, the total distance covered in each direction is the same.

Thus, the **true statement** is:  
- **The two paths involve walking the same distance.**

This is because both paths result in the same total eastward and northward travel, covering the same amount of ground, even though one path appears more complex due to the extra turns.

Would you like further details or clarifications? Here are five related questions you could explore:

1. How can the Pythagorean theorem be applied to check the straight-line distance between A and B?
2. In what situations would the zig-zag path be practically preferred over the direct path?
3. How does the concept of displacement differ from the distance traveled in this context?
4. Can you generalize the solution to a grid of different dimensions?
5. How would adding diagonal paths affect the total distance traveled?

**Tip:** In grid-based problems like this, always separate the horizontal and vertical components of movement to simplify the analysis!

You are in Manhattan and need to walk from point A to point B. Which path is shorter: the zig-zag path or the single-turn path?

Explore the mathematical comparison between a zig-zag path and a single-turn path in a grid system, like Manhattan city blocks. Both paths cover the same total distance by breaking down the movement into horizontal and vertical components. Suitable for Grades 9-12.

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