Question

provide two different degrees of rotation less than 75° but greater than 0° that will turn a regular dodecagon onto itself. (2 points)
a regular dodecagon will turn onto itself after a

and

rotation.

Explanation:

Step1: Recall rotation - symmetry formula

For a regular \(n\) - sided polygon, the angle of rotation \(\theta\) that maps the polygon onto itself is given by \(\theta=\frac{360^{\circ}}{k}\), where \(k\) is a positive integer and \(k\leq n\). For a dodecagon, \(n = 12\), so \(\theta=\frac{360^{\circ}}{k}\), \(k\in\mathbb{Z},1\leq k\leq12\).

Step2: Find valid values of \(k\)

We want \(0<\theta<75^{\circ}\). Substituting \(\theta=\frac{360^{\circ}}{k}\) into the inequality \(0 < \frac{360^{\circ}}{k}<75^{\circ}\). First, \(\frac{360^{\circ}}{k}>0\) is true for \(k>0\). Second, \(\frac{360^{\circ}}{k}<75^{\circ}\), then \(360 < 75k\), so \(k>\frac{360}{75}=4.8\). Since \(k\) is a positive - integer, when \(k = 5\), \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\), when \(k = 6\), \(\theta=\frac{360^{\circ}}{6}=60^{\circ}\).

Answer:

60, 72