Question

which of the following transformations carry this regular polygon onto itself? reflection across ( ell_1 ) rotation of ( 72^{circ} ) clockwise reflection across ( ell_2 ) rotation of ( 45^{circ} ) counterclockwise

Explanation:

To determine which transformations carry the regular polygon (a regular octagon, since the central angle for a regular \( n \)-gon is \( \frac{360^\circ}{n} \), and \( \frac{360^\circ}{8} = 45^\circ \)) onto itself, we analyze each option:

1. Reflection across \( \boldsymbol{\ell_1} \)

A regular octagon has reflection symmetry over lines that either pass through the midpoints of opposite sides or through opposite vertices. If \( \ell_1 \) is a line of symmetry (e.g., passing through the midpoint of a side and the midpoint of the opposite side), reflecting across \( \ell_1 \) will map the octagon onto itself.

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

For a regular \( n \)-gon, the smallest angle of rotational symmetry is \( \frac{360^\circ}{n} \). For an octagon (\( n = 8 \)), the smallest rotational symmetry angle is \( \frac{360^\circ}{8} = 45^\circ \). A rotation of \( 72^\circ \) is not a multiple of \( 45^\circ \) (since \( 72^\circ \div 45^\circ = 1.6 \), not an integer), so this rotation will not map the octagon onto itself.

3. Reflection across \( \boldsymbol{\ell_2} \)

Similar to \( \ell_1 \), if \( \ell_2 \) is a line of symmetry (e.g., passing through opposite vertices), reflecting across \( \ell_2 \) will map the octagon onto itself.

4. Rotation of \( \boldsymbol{45^\circ} \) counterclockwise

The smallest rotational symmetry angle for a regular octagon is \( 45^\circ \) (since \( \frac{360^\circ}{8} = 45^\circ \)). A rotation of \( 45^\circ \) is the smallest angle of rotational symmetry, so this rotation will map the octagon onto itself (as \( 45^\circ \) is a multiple of the smallest rotational symmetry angle).

Correct Options

• Reflection across \( \ell_1 \) (valid, as \( \ell_1 \) is a line of symmetry).
• Reflection across \( \ell_2 \) (valid, as \( \ell_2 \) is a line of symmetry).
• Rotation of \( 45^\circ \) counterclockwise (valid, as \( 45^\circ \) is the rotational symmetry angle for an octagon).

The rotation of \( 72^\circ \) clockwise is invalid (not a multiple of \( 45^\circ \)).

Answer:

• reflection across \( \ell_1 \)
• reflection across \( \ell_2 \)
• rotation of \( 45^\circ \) counterclockwise

(The option "rotation of \( 72^\circ \) clockwise" is incorrect and should not be selected.)