Question

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

Explanation:

To determine which transformations carry the regular polygon (a square, since it has 4 equal sides/angles) onto itself, we analyze each option:

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

The line \( \ell \) is the vertical line of symmetry through the midpoints of the top and bottom sides. Reflecting a square across its vertical line of symmetry maps it onto itself (each point on one side of \( \ell \) has a mirror image on the other side, preserving the square’s shape).

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

For a regular \( n \)-gon, the smallest angle that maps it onto itself under rotation is \( \frac{360^\circ}{n} \). For a square (\( n = 4 \)), this angle is \( \frac{360^\circ}{4} = 90^\circ \). A \( 72^\circ \) rotation is not a multiple of \( 90^\circ \), so it does not map the square onto itself.

3. Rotation of \( \boldsymbol{90^\circ} \) counterclockwise:

A square has rotational symmetry of order 4, meaning rotating it by \( 90^\circ \) (or multiples of \( 90^\circ \)) about its center maps it onto itself. A \( 90^\circ \) counterclockwise rotation aligns the square’s vertices with their original positions (e.g., top-right vertex moves to top-left, etc.).

4. Rotation of \( \boldsymbol{120^\circ} \) counterclockwise:

\( 120^\circ \) is not a multiple of \( 90^\circ \) (the square’s rotational symmetry angle). Thus, a \( 120^\circ \) rotation does not map the square onto itself.

Correct Options:

• Reflection across \( \ell \) (valid, due to line symmetry).
• Rotation of \( 90^\circ \) counterclockwise (valid, due to rotational symmetry).

(Note: The original checked options included incorrect ones; the corrected valid transformations are reflection across \( \ell \) and rotation of \( 90^\circ \) counterclockwise.)

Answer:

• reflection across \( \ell \)
• rotation of \( 90^\circ \) counterclockwise