Question

1. regular pentagon qrstu is shown below. segments have been drawn from each vertex, perpendicular to the opposite side. intersection points are marked as shown. all segments intersect at center point o. (a) list all angles of rotational symmetry about point o for this pentagon. (b) what counterclockwise rotation about point o would be needed to map point s onto point q? (c) what point would each of the following get mapped to if reflected across \\( \overline{qa} \\)?

Explanation:

(a)

Step1: Calculate the smallest angle of rotational symmetry

For a regular \(n -\)gon, the smallest angle of rotational symmetry \(\theta=\frac{360^{\circ}}{n}\). Here \(n = 5\), so \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\).

Step2: List all angles of rotational symmetry

The angles of rotational symmetry are \(k\times72^{\circ}\), where \(k = 1,2,3,4,5\). So the angles are \(72^{\circ},144^{\circ},216^{\circ},288^{\circ},360^{\circ}\)

(b)

Step1: Count the number of intervals between \(S\) and \(Q\)

Counting the number of intervals (each interval is \(72^{\circ}\)) from \(S\) to \(Q\) counter - clockwise. There are \(3\) intervals.

Step2: Calculate the rotation angle

Since each interval is \(72^{\circ}\), the rotation angle \(\alpha=3\times72^{\circ}=216^{\circ}\)

Answer:

(a) The angles of rotational symmetry are \(72^{\circ},144^{\circ},216^{\circ},288^{\circ},360^{\circ}\)
(b) A counter - clockwise rotation of \(216^{\circ}\) about point \(O\) is needed to map point \(S\) onto point \(Q\)