The question is asking which transformations will carry this regular hexagon onto itself. For a regular hexagon, there are several transformations that can map it back onto itself, including specific rotations and reflections.

Explanation of each option:

1. Rotation of 72° clockwise: Since 72° is not a multiple of 60°, it would not map a regular hexagon onto itself, as the angles between each side of a regular hexagon are 60°.

2. Rotation of 60° clockwise: This is valid. A rotation of 60° (or any multiple of 60°) will map the hexagon onto itself since it has six-fold rotational symmetry.

3. Reflection across line $l$: If line $l$ is an axis of symmetry through opposite vertices or midpoints of opposite sides, then a reflection across this line would map the hexagon onto itself.

4. Rotation of 60° counterclockwise: This is also valid for the same reason as option 2; rotating by 60° counterclockwise will map the hexagon onto itself.

Correct Answers:

The transformations that map the hexagon onto itself are:

• Rotation of 60° clockwise
• Reflection across line $l$ (assuming line $l$ is a symmetry axis)
• Rotation of 60° counterclockwise

Let me know if you need further details or explanations!

Related Questions:

1. What are the symmetries of a regular hexagon?
2. How do rotational symmetries differ between polygons with different numbers of sides?
3. Why does a regular hexagon have six axes of symmetry?
4. What other transformations map regular polygons onto themselves?
5. How can you determine rotational symmetry for any regular polygon?

Tip:

To find rotational symmetries of a regular polygon, divide 360° by the number of sides to find the smallest rotation angle that maps the shape onto itself.