Question

which of the following transformations carry this regular polygon onto itself?
rotation of 30° counterclockwise
rotation of 90° counterclockwise
rotation of 120° counterclockwise
rotation of 72° clockwise

Explanation:

Step1: Find the order of rotational symmetry

A square has 4 sides. The formula for the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For a square, \(n = 4\), so \(\frac{360^{\circ}}{4}=90^{\circ}\).

Step2: Check each rotation

• For a rotation of \(30^{\circ}\): \(30^{\circ}\) is not a multiple of \(90^{\circ}\), so it will not map the square onto itself.
• For a rotation of \(90^{\circ}\): Since \(90^{\circ}\) is the angle of rotational symmetry of a square, a \(90^{\circ}\) counter - clockwise rotation will map the square onto itself.
• For a rotation of \(120^{\circ}\): \(120^{\circ}\) is not a multiple of \(90^{\circ}\), so it will not map the square onto itself.
• For a rotation of \(72^{\circ}\): \(72^{\circ}\) is not a multiple of \(90^{\circ}\), so it will not map the square onto itself.

Answer:

rotation of \(90^{\circ}\) counterclockwise