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The problem involves a reflection of triangle \( XYZ \) across the line \( y = x \), resulting in triangle \( X'Y'Z' \). Here’s how we can break it down:

### Part (a): Filling in the blanks for the coordinates
The arrows in part (a) show the transformation of points \( X \), \( Y \), and \( Z \) into \( X' \), \( Y' \), and \( Z' \) respectively. The original and final coordinates are partially given.

The transformation rule for reflecting across the line \( y = x \) swaps the \( x \)- and \( y \)-coordinates of each point. This means:
- \( (x, y) \rightarrow (y, x) \)

Now, applying this rule:
1. \( X(-2, -6) \) transforms to \( X'(-6, -2) \)
2. \( Y(6, 2) \) transforms to \( Y'(2, 6) \)
3. \( Z(5, -3) \) transforms to \( Z'(-3, 5) \)

The highlighted answers are already correct.

### Part (b): Choosing the general rule for the reflection
Since the reflection across the line \( y = x \) swaps the coordinates, the general rule is:
- \( (x, y) \rightarrow (y, x) \)

Thus, the correct option is \( (x, y) \rightarrow (y, x) \), which corresponds to the second choice.

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Let me know if you'd like further clarification, and here are some additional questions to consider:

1. What is the general rule for reflecting a point across the x-axis?
2. How would the reflection change if the triangle was reflected over the y-axis instead?
3. Can you describe the transformation if the reflection was over the line \( y = -x \)?
4. How would the coordinates of point \( X'(-6, -2) \) change if it was reflected back over the line \( y = x \)?
5. What geometric properties remain unchanged during a reflection?

**Tip:** Reflections over lines like \( y = x \) or the coordinate axes always maintain the distance between points, preserving the size and shape of geometric figures.

Triangle XYZ is reflected across the line y = x. The result is triangle X'Y'Z'. What are the new coordinates after the reflection and the general rule describing this transformation?

Understand how to reflect triangle XYZ across the line y = x and determine the new coordinates of the vertices. This problem involves coordinate geometry concepts, reflections, and the transformation rule (x, y) → (y, x). Suitable for students in Grades 8-10.

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