To solve this problem, let's break down the transformations and analyze which vertices, if any, remain invariant:

1. Reflection in the line $x = 4$:

   • The reflection line $x = 4$ is a vertical line through the center of the square.
   • When each vertex is reflected over this line:
     • Vertex $A$ at $(2, 6)$ reflects to $(6, 6)$, moving to the right.
     • Vertex $B$ at $(6, 6)$ reflects to $(2, 6)$, moving to the left.
     • Vertex $D$ at $(2, 2)$ reflects to $(6, 2)$.
     • Vertex $C$ at $(6, 2)$ reflects to $(2, 2)$.

   After the reflection, the vertices switch places as follows:

   • $A \leftrightarrow B$
   • $C \leftrightarrow D$

2. 90° Anticlockwise Rotation about $(4, 4)$:

   • Now we apply a 90° anticlockwise rotation around the point $(4, 4)$.
   • For each vertex after the reflection:
     • Reflected $A$ (now at $(6, 6)$) moves to $(2, 6)$.
     • Reflected $B$ (now at $(2, 6)$) moves to $(2, 2)$.
     • Reflected $D$ (now at $(6, 2)$) moves to $(6, 6)$.
     • Reflected $C$ (now at $(2, 2)$) moves to $(6, 2)$.

After completing both transformations, we observe that no vertex remains in its original position, meaning no vertices are invariant under these two transformations.

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How does the location of the center of rotation affect the final positions of the vertices?
2. What would happen if the rotation was clockwise instead of anticlockwise?
3. How would the solution change if we used a different line for reflection?
4. Can we find invariant points under a sequence of reflections and rotations in other shapes?
5. What is the significance of an invariant point in transformations?

Tip: When performing multiple transformations, it’s often helpful to apply each one step-by-step, tracking each point's movement to understand the final configuration.