Question

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

Explanation:

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

1. Reflection across \( \boldsymbol{\ell} \)

A regular hexagon has reflectional symmetry. The line \( \ell \) appears to pass through a vertex and the midpoint of the opposite side (or two vertices, given the markings). Reflecting across such a line maps the hexagon onto itself, so this transformation is valid.

2. Rotation of \( \boldsymbol{60^\circ} \) counterclockwise

A regular hexagon has rotational symmetry of order 6 (since \( 360^\circ \div 6 = 60^\circ \)). Rotating by \( 60^\circ \) (a multiple of \( 60^\circ \)) maps the hexagon onto itself. Thus, this transformation is valid.

3. Rotation of \( \boldsymbol{72^\circ} \) clockwise

A regular hexagon’s rotational symmetry is based on \( 60^\circ \) (not \( 72^\circ \), which is for a regular pentagon: \( 360^\circ \div 5 = 72^\circ \)). Rotating by \( 72^\circ \) does not map a hexagon onto itself, so this is invalid.

4. Rotation of \( \boldsymbol{72^\circ} \) counterclockwise

Similar to the previous case, \( 72^\circ \) is not a rotational symmetry angle for a hexagon (which requires multiples of \( 60^\circ \)). Thus, this is invalid.

Correct Options:

• A. reflection across \( \ell \)
• B. rotation of \( 60^\circ \) counterclockwise

(Note: The original checked options included errors; the valid transformations for a regular hexagon are reflection across symmetry lines and rotations by \( 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ \), etc.)

Answer:

A. reflection across \( \ell \)
B. rotation of \( 60^\circ \) counterclockwise