Question

1. the regular polygon below is rotated about its center. which angle of rotation will carry the figure onto itself? a) 540° b) 60° c) 216° d) 108°

Explanation:

Step1: Find the formula for rotation angle of a regular polygon

The formula for the smallest positive angle of rotation that maps a regular \(n -\)sided polygon onto itself is \(\frac{360^{\circ}}{n}\). For a regular pentagon (\(n = 5\)), the smallest positive rotation angle is \(\frac{360^{\circ}}{5}=72^{\circ}\). Any multiple of \(72^{\circ}\) will also map the pentagon onto itself.

Step2: Check each option

• Option A: \(540^{\circ}\div72^{\circ}=7.5\), not an integer.
• Option B: \(60^{\circ}\div72^{\circ}=\frac{5}{6}\), not an integer.
• Option C: \(216^{\circ}\div72^{\circ}=3\), an integer.
• Option D: \(108^{\circ}\div72^{\circ}=1.5\), not an integer.

Answer:

C. \(216^{\circ}\)