Question

which of the following transformations carry this regular polygon onto itself?
rotation of 90° clockwise
reflection across ℓ
rotation of 120° counterclockwise
rotation of 60° counterclockwise

Explanation:

1. Regular Polygon (Square) Properties: A square has rotational symmetry of order 4, meaning it maps onto itself under rotations of \(90^\circ\), \(180^\circ\), \(270^\circ\), etc. It also has reflectional symmetry across horizontal, vertical, and diagonal lines.
2. Analyzing Each Option:

• Rotation of \(90^\circ\) clockwise: A square rotated \(90^\circ\) clockwise aligns with its original position (rotational symmetry), so this works.
• Reflection across \(\boldsymbol{\ell}\): The line \(\ell\) is horizontal (midline of the square). Reflecting a square across its horizontal midline maps it onto itself (reflectional symmetry), so this works.
• Rotation of \(120^\circ\) counterclockwise: A square’s rotational symmetry angles are multiples of \(90^\circ\) (\(90^\circ, 180^\circ, 270^\circ\)). \(120^\circ\) is not a multiple of \(90^\circ\), so this does not map the square onto itself.
• Rotation of \(60^\circ\) counterclockwise: Similarly, \(60^\circ\) is not a multiple of \(90^\circ\), so this does not map the square onto itself.

Answer:

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