Question

sc 4: i can determine reflectional and rotational symmetries of
4a) this is a regular dodecagon (12 - sided figure).
what angles of rotation between 0° and 360° will map this
shape onto itself?

Explanation:

Step1: Calculate the minimum rotation angle

For a regular \(n -\)sided polygon, the minimum positive rotation angle \(\theta\) that maps the polygon onto itself is \(\theta=\frac{360^{\circ}}{n}\). Here \(n = 12\), so \(\theta=\frac{360^{\circ}}{12}=30^{\circ}\).

Step2: Find all rotation angles

The rotation angles \(\alpha\) between \(0^{\circ}\) and \(360^{\circ}\) that map the regular dodecagon onto itself are of the form \(\alpha = k\times\frac{360^{\circ}}{12}\), where \(k = 1,2,\cdots,11\).
When \(k = 1\), \(\alpha=30^{\circ}\); when \(k = 2\), \(\alpha = 60^{\circ}\); when \(k=3\), \(\alpha=90^{\circ}\); when \(k = 4\), \(\alpha = 120^{\circ}\); when \(k=5\), \(\alpha=150^{\circ}\); when \(k = 6\), \(\alpha=180^{\circ}\); when \(k=7\), \(\alpha=210^{\circ}\); when \(k = 8\), \(\alpha=240^{\circ}\); when \(k=9\), \(\alpha=270^{\circ}\); when \(k = 10\), \(\alpha=300^{\circ}\); when \(k = 11\), \(\alpha=330^{\circ}\)

Answer:

\(30^{\circ},60^{\circ},90^{\circ},120^{\circ},150^{\circ},180^{\circ},210^{\circ},240^{\circ},270^{\circ},300^{\circ},330^{\circ}\)