A line segment has endpoints at (3, 2) and (2, –3). Which reflection will produce an image with endpoints at (3, –2) and (2, 3)?

a reflection of the line segment across the *x*-axis

a reflection of the line segment across the *y*-axis

a reflection of the line segment across the line *y* = *x*

a reflection of the line segment across the line *y* = –*x*

To determine which reflection will produce the desired image, let's analyze the problem by reflecting the given endpoints across each of the specified lines.

### Original Endpoints:
- Point A: (3, 2)
- Point B: (2, -3)

### Desired Image Endpoints:
- Point A': (3, -2)
- Point B': (2, 3)

### Reflection Analysis:

1. **Reflection across the *x*-axis**:
   - Reflection rule: (*x*, *y*) → (*x*, -*y*)
   - Point A (3, 2) → (3, -2)
   - Point B (2, -3) → (2, 3)
   - This reflection results in the desired endpoints, so this is the correct reflection.

2. **Reflection across the *y*-axis**:
   - Reflection rule: (*x*, *y*) → (-*x*, *y*)
   - Point A (3, 2) → (-3, 2)
   - Point B (2, -3) → (-2, -3)
   - This does not match the desired endpoints.

3. **Reflection across the line *y* = *x***:
   - Reflection rule: (*x*, *y*) → (*y*, *x*)
   - Point A (3, 2) → (2, 3)
   - Point B (2, -3) → (-3, 2)
   - This does not match the desired endpoints.

4. **Reflection across the line *y* = -*x***:
   - Reflection rule: (*x*, *y*) → (-*y*, -*x*)
   - Point A (3, 2) → (-2, -3)
   - Point B (2, -3) → (3, -2)
   - This does not match the desired endpoints.

### Conclusion:
The correct reflection that will produce the image with endpoints at (3, -2) and (2, 3) is **a reflection of the line segment across the *x*-axis**.

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Would you like further details or have any questions about this? Here are some related questions to explore:

1. What are the general reflection rules for coordinates in a 2D plane?
2. How does reflecting across the origin differ from reflecting across the *x*- or *y*-axis?
3. What would be the new coordinates if we reflect the segment across the line *y* = *x* and then across the *x*-axis?
4. How can you determine the line of reflection if you are given the original and reflected coordinates?
5. What other transformations, besides reflection, can be used to obtain specific coordinates?
6. How does reflecting a shape across different axes affect its orientation and position?
7. Can reflection across a line other than the axes or *y* = *x* be used to obtain desired coordinates?
8. What happens to the slope of a line segment when it is reflected across different lines?

**Tip:** Visualizing transformations like reflections by sketching on graph paper or using graphing software can help in understanding the changes in coordinates.

Explore how to reflect a line segment across different axes and lines in the coordinate plane. This problem involves determining the correct reflection that will produce specific endpoint coordinates. Detailed step-by-step analysis and solution provided.

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