Question

which of the following transformations carry this regular polygon onto itself? reflection across l rotation of 45° counterclockwise rotation of 60° clockwise rotation of 72° counterclockwise

Explanation:

Step1: Reflection property

A regular polygon has reflectional symmetry. If the line \(l\) is a line of symmetry (which it is for a regular hexagon as shown in the figure), then reflection across \(l\) will map the polygon onto itself.

Step2: Rotational symmetry formula

The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\theta=\frac{360^{\circ}}{n}\). For a hexagon \(n = 6\), so \(\theta=\frac{360^{\circ}}{6}=60^{\circ}\). A rotation of \(60^{\circ}\) (clock - wise or counter - clockwise) will map the regular hexagon onto itself. A rotation of \(45^{\circ}\) (since \(45^{\circ}\) does not divide \(360^{\circ}\) evenly for \(n = 6\)) and \(72^{\circ}\) (since \(72^{\circ}\) does not divide \(360^{\circ}\) evenly for \(n=6\)) will not map the hexagon onto itself.

Answer:

reflection across \(l\), rotation of \(60^{\circ}\) clockwise