Question

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

Explanation:

Step1: Identify the polygon

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

Step2: Analyze reflection

The line \( l \) is a line of symmetry (since it passes through a vertex and the mid - point of the opposite side in a regular pentagon). So reflection across line \( l \) will carry the pentagon onto itself.

Step3: Analyze other rotations

• For a rotation of \( 90^{\circ} \): Since the rotational symmetry angle is \( 72^{\circ} \), \( 90^{\circ} \) is not a multiple of \( 72^{\circ} \) (except \( 0^{\circ} \) and multiples of \( 72^{\circ} \) for a pentagon), so a rotation of \( 90^{\circ} \) counterclockwise will not carry the pentagon onto itself.
• For a rotation of \( 120^{\circ} \): \( 120^{\circ} \) is not a multiple of \( 72^{\circ} \), so a rotation of \( 120^{\circ} \) counterclockwise will not carry the pentagon onto itself.

Answer:

B. reflection across \( l \)
C. rotation of \( 72^{\circ} \) counterclockwise

(Here we assume the options are labeled as follows: A. rotation of \( 90^{\circ} \) counterclockwise, B. reflection across \( l \), C. rotation of \( 72^{\circ} \) counterclockwise, D. rotation of \( 120^{\circ} \) counterclockwise)