Question

which of the following transformations carry this regular polygon onto itself?
rotation of 72° counterclockwise
rotation of 90° counterclockwise
rotation of 72° clockwise
rotation of 60° counterclockwise

Explanation:

Step1: Determine the polygon type

The figure is a regular pentagon. For a regular \( n \)-sided polygon, the angle of rotational symmetry is given by \( \frac{360^\circ}{n} \). For a pentagon, \( n = 5 \), so the rotational symmetry angle is \( \frac{360^\circ}{5}=72^\circ \).

Step2: Analyze each rotation

• Rotation of \( 72^\circ \) counterclockwise: Since \( 72^\circ \) is the rotational symmetry angle for a regular pentagon, rotating \( 72^\circ \) counterclockwise will map the pentagon onto itself.
• Rotation of \( 90^\circ \) counterclockwise: \( 90^\circ \) is not a multiple of \( 72^\circ \), so this rotation will not map the pentagon onto itself.
• Rotation of \( 72^\circ \) clockwise: Rotating \( 72^\circ \) clockwise is equivalent to rotating \( - 72^\circ \) counterclockwise, and since \( 72^\circ \) is the rotational symmetry angle, this will map the pentagon onto itself.
• Rotation of \( 60^\circ \) counterclockwise: \( 60^\circ \) is not a multiple of \( 72^\circ \), so this rotation will not map the pentagon onto itself.

Answer:

A. rotation of \( 72^\circ \) counterclockwise
C. rotation of \( 72^\circ \) clockwise