The question asks which transformations will carry a regular polygon (square) onto itself. The square has a diagonal labeled as line $\ell$, indicating a potential reflection axis.

Analysis of Each Transformation:

1. Rotation of 90° Clockwise: A square rotated by 90° will map onto itself because all sides and angles are symmetrical.

2. Rotation of 90° Counterclockwise: Similarly, a 90° counterclockwise rotation will also map the square onto itself.

3. Rotation of 45° Clockwise: This rotation would not map the square onto itself since it doesn’t align with the square's symmetry points (90° rotations do).

4. Reflection across line $\ell$: Since line $\ell$ is a diagonal of the square, reflecting the square across this line will also map it onto itself, as squares have reflective symmetry along both diagonals.

Conclusion:

The correct transformations are:

• Rotation of 90° Clockwise
• Rotation of 90° Counterclockwise
• Reflection across line $\ell$

Would you like further details on why each transformation does or does not work?

Related Questions

1. What other types of transformations can map a square onto itself?
2. How does symmetry play a role in determining transformations for regular polygons?
3. Can irregular polygons have similar transformations that map them onto themselves?
4. How would these transformations apply to polygons with more sides, like a hexagon?
5. Why does a 45° rotation not work for a square but work for an octagon?

Tip

For any regular polygon, rotations by multiples of $\frac{360°}{\text{number of sides}}$ and reflections across symmetry axes will always map the shape onto itself.