Question

determining the order of rotational symmetry
consider the regular dodecagon. which statements are true regarding the regular dodecagon? check all that apply.
the smallest angle of rotational symmetry for the dodecagon is 30°.
the dodecagon has a rotational symmetry of 180°.
the order of rotational symmetry for the dodecagon is 10.
the dodecagon has a rotational symmetry of 135°.
the angles of rotational symmetry for the dodecagon are multiples of 30°.

Explanation:

Step1: Calculate the smallest angle of rotational symmetry

The formula for the smallest angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a dodecagon (\(n = 12\)), \(\frac{360^{\circ}}{12}=30^{\circ}\).

Step2: Check if \(180^{\circ}\) is a rotational symmetry angle

Since \(180\div30 = 6\) (an integer), \(180^{\circ}\) is a multiple of \(30^{\circ}\). So the dodecagon has a rotational symmetry of \(180^{\circ}\).

Step3: Check the order of rotational symmetry

The order of rotational symmetry of a regular \(n -\)sided polygon is \(n\). For a dodecagon (\(n = 12\)), the order is \(12\), not \(10\).

Step4: Check if \(135^{\circ}\) is a rotational symmetry angle

Since \(135\div30=4.5\) (not an integer), \(135^{\circ}\) is not a multiple of \(30^{\circ}\). So the dodecagon does not have a rotational symmetry of \(135^{\circ}\).

Step5: Check the nature of rotational - symmetry angles

Since the smallest angle of rotational symmetry is \(30^{\circ}\), the angles of rotational symmetry are \(k\times30^{\circ}\), where \(k = 1,2,\cdots,12\). So the angles of rotational symmetry for the dodecagon are multiples of \(30^{\circ}\).

Answer:

The smallest angle of rotational symmetry for the dodecagon is \(30^{\circ}\); The dodecagon has a rotational symmetry of \(180^{\circ}\); The angles of rotational symmetry for the dodecagon are multiples of \(30^{\circ}\).