The image shows a square centered at the origin on a coordinate plane. To determine the transformation that carries the square onto itself, we need to identify any symmetry or rotational properties of the square.

There are several transformations that could map the square onto itself:

1. Rotation: A square centered at the origin is symmetric under 90°, 180°, 270°, and 360° rotations about the origin.
2. Reflection: The square is symmetric under reflection across the x-axis, y-axis, and the lines $y = x$ and $y = -x$.
3. Translation: Since the square is centered at the origin, a translation would not map the square onto itself unless it is a translation by 0 units (no movement).
4. Identity transformation: This is the transformation where the square remains unchanged.

The most common transformation that carries the square onto itself:

• Rotation by 90°, 180°, 270°, or 360° around the origin.
• Reflection across the axes or the lines $y = x$ and $y = -x$.

Would you like a detailed explanation of any of these transformations?

Here are five relative questions:

1. What is the difference between rotation and reflection symmetries for polygons?
2. How can we determine the angle of rotation that maps a polygon onto itself?
3. What types of polygons have rotational symmetry, and why?
4. How can reflections across axes be expressed algebraically in coordinate transformations?
5. How does the number of sides of a polygon affect its symmetry properties?

Tip: Always check if a figure is centered at the origin, as it simplifies many transformations like rotations and reflections.