Question

which of the following transformations carry this regular polygon onto itself? rotation of 30° clockwise rotation of 72° counterclockwise rotation of 90° counterclockwise reflection across l

Explanation:

Step1: Analyze the regular polygon

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 \( n = 5 \), the rotational symmetry 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 \( 30^\circ \) clockwise: \( 30^\circ \) is not a multiple of \( 72^\circ \), so this rotation will not map the pentagon onto itself.
• Rotation of \( 72^\circ \) counterclockwise: Since \( 72^\circ=\frac{360^\circ}{5} \), rotating the pentagon by \( 72^\circ \) (or any multiple of \( 72^\circ \)) will map it 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.
• Reflection across \( l \): Line \( l \) passes through a vertex and the midpoint of the opposite side (as seen from the diagram, it splits the bottom side into two equal parts and goes through the top vertex), so reflecting across \( l \) will map the pentagon onto itself.

Answer:

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