Question

which of the following transformations carry this regular polygon onto itself? image of a regular pentagon with a line of reflection l rotation of 90° counterclockwise reflection across l rotation of 72° counterclockwise rotation of 120° counterclockwise

Explanation:

Step1: Analyze the regular polygon

The figure is a regular pentagon. A regular \( n \)-sided polygon has rotational symmetry of order \( n \), meaning the smallest angle of rotation that maps it onto itself is \( \frac{360^\circ}{n} \). For a pentagon, \( n = 5 \), so the smallest rotational angle is \( \frac{360^\circ}{5}=72^\circ \). Also, a regular pentagon has reflection symmetry across lines that pass through a vertex and the midpoint of the opposite side (like line \( l \) here).

Step2: Evaluate each transformation

• Rotation of \( 90^\circ \) counterclockwise: The smallest rotational angle for a pentagon is \( 72^\circ \), and \( 90^\circ \) is not a multiple of \( 72^\circ \) (since \( 90\div72 = 1.25 \)), so this rotation will not map the pentagon onto itself.
• Reflection across \( l \): Line \( l \) passes through a vertex and the midpoint of the opposite side, which is a line of reflection symmetry for a regular pentagon. So reflecting across \( l \) will map the pentagon onto itself.
• Rotation of \( 72^\circ \) counterclockwise: Since the smallest rotational angle for a regular pentagon is \( 72^\circ \), rotating by \( 72^\circ \) (a multiple of \( 72^\circ \)) will map the pentagon onto itself.
• Rotation of \( 120^\circ \) counterclockwise: \( 120\div72=\frac{5}{3}\), which is not an integer, so \( 120^\circ \) is not a multiple of the smallest rotational angle, and this rotation will not map the pentagon onto itself.

Answer:

reflection across \( l \), rotation of \( 72^\circ \) counterclockwise