https://file.aiquickdraw.com/mathbot/file/17256056646827wt8duzl.jpg

The image shows a coordinate plane with points and lines, and the task asks for the coordinates of a reflection over the line \(y = x\). Let's go step by step to reflect the vertices of the green polygon over the line \(y = x\).

### Steps to reflect over the line \(y = x\):
1. **Swap the coordinates** of each point \((x, y)\). After reflection, the point \((x, y)\) becomes \((y, x)\).

### Coordinates of the vertices before reflection:
Let's extract the coordinates of the vertices from the graph:
- Vertex 1: \((-3, 4)\)
- Vertex 2: \((-1, 5)\)
- Vertex 3: \((2, 2)\)

### Reflecting these points over the line \(y = x\):
- Reflect \((-3, 4)\): This becomes \((4, -3)\).
- Reflect \((-1, 5)\): This becomes \((5, -1)\).
- Reflect \((2, 2)\): This stays the same \((2, 2)\) because it's on the line \(y = x\).

### Final coordinates after reflection:
1. \((4, -3)\)
2. \((5, -1)\)
3. \((2, 2)\)

Would you like more details or further clarification on any part of this? Here are some related questions you might find useful:

1. How do we reflect a point over the x-axis or y-axis?
2. What happens if a point lies on the line of reflection?
3. Can you explain how to find the equation of the line of reflection for any two given points and their reflections?
4. What transformations preserve distances between points?
5. How do other types of reflections (e.g., over arbitrary lines) differ from reflections over \(y = x\)?

**Tip**: Reflections maintain the shape and size of objects, but they flip their orientation across the line of reflection.

Write the coordinates of the vertices after a reflection over the line y = x.

Learn how to reflect vertices of a polygon over the line y = x. This coordinate geometry problem demonstrates reflections and symmetry, with step-by-step solutions for each point. Suitable for students in Grades 6-8.

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