Question

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

Explanation:

Step1: Find the angle of rotational symmetry

The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a pentagon (\(n = 5\)), \(\frac{360^{\circ}}{5}=72^{\circ}\).

Step2: Check each rotation

• For a rotation of \(90^{\circ}\) (counter - clockwise or clockwise): \(90\div72=\frac{5}{4}\), not an integer. So, a \(90^{\circ}\) rotation does not map the pentagon onto itself.
• For a rotation of \(120^{\circ}\) (counter - clockwise): \(120\div72=\frac{5}{3}\), not an integer. So, a \(120^{\circ}\) rotation does not map the pentagon onto itself.
• For a rotation of \(72^{\circ}\) (clockwise): \(72\div72 = 1\). So, a \(72^{\circ}\) clockwise rotation maps the pentagon onto itself.

Answer:

rotation of \(72^{\circ}\) clockwise