Question

consider the original squares side lengths. do the side lengths remain the same after: any horizontal or vertical reflection? any horizontal or vertical translation? any rotation?

Explanation:

Any horizontal or vertical reflection

Reflection is a rigid transformation. Rigid transformations preserve the shape and size of a figure. Since a square's side - length is a measure of its size, after a horizontal or vertical reflection, the side - lengths of the square remain the same.

Any horizontal or vertical translation

Translation is also a rigid transformation. It moves the square without changing its shape or size. So, the side - lengths of the square do not change after a horizontal or vertical translation.

Any rotation

Rotation is a rigid transformation. When a square is rotated, its orientation changes, but its shape and size (including side - lengths) remain unchanged.

Answer:

• Any horizontal or vertical reflection: Yes
• Any horizontal or vertical translation: Yes
• Any rotation: Yes