Question

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

Explanation:

Step1: Find the rotational symmetry of a square

A square has \(n = 4\) sides. The formula for the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\). For \(n = 4\), \(\frac{360^{\circ}}{4}=90^{\circ}\). A square maps onto itself when rotated by an angle that is a multiple of \(90^{\circ}\).

Step2: Analyze each option

• For a rotation of \(72^{\circ}\): \(72^{\circ}\) is not a multiple of \(90^{\circ}\) (\(72\div90 = 0.8\)).
• For a rotation of \(120^{\circ}\): \(120^{\circ}\) is not a multiple of \(90^{\circ}\) (\(120\div90=\frac{4}{3}\)).
• For a rotation of \(30^{\circ}\): \(30^{\circ}\) is not a multiple of \(90^{\circ}\) (\(30\div90=\frac{1}{3}\)).
• For a rotation of \(90^{\circ}\): \(90^{\circ}\) is a multiple of \(90^{\circ}\) (\(90\div90 = 1\)).

Answer:

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