Question

which of the following transformations carry this regular polygon onto itself?
rotation of 60° counterclockwise
rotation of 30° clockwise
rotation of 72° counterclockwise
reflection across ℓ

Explanation:

To determine which transformations carry a regular pentagon (the given polygon) onto itself, we analyze each option:

1. Rotation of \( 60^\circ \) counterclockwise

A regular pentagon has a rotational symmetry of \( \frac{360^\circ}{5} = 72^\circ \). A \( 60^\circ \) rotation is not a multiple of \( 72^\circ \), so this does not map the pentagon onto itself.

2. Rotation of \( 30^\circ \) clockwise

Again, \( 30^\circ \) is not a multiple of \( 72^\circ \), so this rotation does not preserve the pentagon’s symmetry.

3. Rotation of \( 72^\circ \) counterclockwise

The rotational symmetry of a regular \( n \)-gon is \( \frac{360^\circ}{n} \). For \( n = 5 \) (pentagon), \( \frac{360^\circ}{5} = 72^\circ \). Thus, a \( 72^\circ \) rotation (or any multiple of \( 72^\circ \)) maps the pentagon onto itself.

4. Reflection across line \( \ell \)

The line \( \ell \) is a line of symmetry (it bisects a side and passes through the opposite vertex, as seen in the diagram with congruent marked segments). Reflecting a regular pentagon over its line of symmetry maps it onto itself.

Correct Options:

• Rotation of \( 72^\circ \) counterclockwise
• Reflection across \( \ell \)

(Note: The initial "checked" marks for \( 60^\circ \) and \( 30^\circ \) rotations were incorrect; those rotations do not preserve the pentagon’s symmetry.)

Answer:

C. rotation of \( 72^\circ \) counterclockwise, D. reflection across \( \ell \)

(Assuming options are labeled as: A. rotation of \( 60^\circ \) counterclockwise, B. rotation of \( 30^\circ \) clockwise, C. rotation of \( 72^\circ \) counterclockwise, D. reflection across \( \ell \))