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.

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.