Let's analyze the given problem step by step:

(a) Fill in the blanks for the coordinates after the reflection:

1. Original coordinates:

   • $X(-2, -6)$
   • $Y(6, 2)$
   • $Z(5, -3)$

   We need to find their reflected counterparts:

   • $X'(-6, -1)$
   • $Y'(2, 6)$
   • $Z'(-3, 5)$

(b) General rule for the reflection:

The problem asks for the rule that describes the reflection mapping $\Delta XYZ$ to $\Delta X'Y'Z'$.

Looking at the transformation of points:

• For $X(-2, -6) \to X'(-6, -1)$
• For $Y(6, 2) \to Y'(2, 6)$
• For $Z(5, -3) \to Z'(-3, 5)$

It seems that the coordinates are being reflected in such a way that the $(x, y)$ coordinates are mapped to $(-y, -x)$.

Thus, the correct general rule is: $(x, y) \to (-y, -x)$

This matches the third option in part (b).

Would you like more details on how the reflection rule was identified? Let me know if you have any other questions or if you'd like further explanation.

Here are 5 related questions:

1. How do different types of reflections (across the x-axis, y-axis, and other lines) affect coordinates?
2. What would happen to the coordinates if reflected across the line $y = x$?
3. Can reflections be combined with other transformations, such as rotations or translations?
4. How do we describe reflections in three-dimensional space?
5. What are the properties of rigid transformations like reflections in geometry?

Tip: Always look for patterns in coordinate transformations by analyzing how individual points are mapped.