Question

which of the following transformations carry this regular polygon onto itself? image of a regular hexagon rotation of 72° counterclockwise rotation of 72° clockwise rotation of 40° counterclockwise rotation of 60° counterclockwise

Explanation:

Step1: Identify the polygon

The figure is a regular hexagon, which has \( n = 6 \) sides.

Step2: Calculate rotational symmetry angle

For a regular \( n \)-sided polygon, the minimum angle of rotation that maps it onto itself is \( \frac{360^\circ}{n} \). For \( n = 6 \), this angle is \( \frac{360^\circ}{6}=60^\circ \). Any multiple of this angle (in both clockwise and counterclockwise directions) will also map the polygon onto itself.

Step3: Analyze each option

• Rotation of \( 72^\circ \) (counterclockwise or clockwise): \( 72^\circ \) is not a multiple of \( 60^\circ \), so this rotation will not map the hexagon onto itself.
• Rotation of \( 40^\circ \) (counterclockwise): \( 40^\circ \) is not a multiple of \( 60^\circ \), so this rotation will not map the hexagon onto itself.
• Rotation of \( 60^\circ \) (counterclockwise): \( 60^\circ \) is the minimum rotational symmetry angle for a hexagon, so this rotation will map the hexagon onto itself.

Answer:

rotation of \( 60^\circ \) counterclockwise