To fully describe the transformation of square ABCD based on the given information:

Given:

• $A(1, 1), B(-1, 3), C(1, 5), D(3, 3)$
• A transformation maps:
  • $A$ to $B$
  • $B$ to $C$
• The only invariant point is $(1, 3)$

Analysis:

1. Rotation Center: The only invariant point is $(1, 3)$. This means that the transformation likely involves a rotation around the point $(1, 3)$.

2. Determining the Angle of Rotation:

   • $A(1, 1)$ is mapped to $B(-1, 3)$, meaning point $A$ rotates to become point $B$.
   • The distance from $A(1, 1)$ to the invariant point $(1, 3)$ is 2 units (since it's vertically aligned).
   • After the rotation, $B(-1, 3)$ is horizontally aligned with the invariant point and also 2 units away, suggesting a 90-degree counterclockwise rotation.

3. Verifying for $B$ to $C$:

   • If we apply the same 90-degree counterclockwise rotation to $B(-1, 3)$, it should map to $C(1, 5)$, which holds true as it moves 2 units above the invariant point.

Conclusion:

The transformation is a 90-degree counterclockwise rotation about the point $(1, 3)$, which is the invariant point.

Do you need further details or have any questions?

Expanding questions:

1. What would the transformation look like if it were a 90-degree clockwise rotation instead?
2. How do you calculate the matrix for a 90-degree counterclockwise rotation?
3. Can you perform a rotation about a point other than the origin mathematically?
4. What other types of transformations could maintain a single invariant point?
5. How does this transformation change the side lengths and angles of the square?

Tip: To confirm a rotation transformation, always check distances between points before and after the transformation to verify the correctness of the angle and center of rotation.