Question

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

Explanation:

Step1: Calculate the rotational symmetry of a regular hexagon

The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a hexagon \(n = 6\), so \(\frac{360^{\circ}}{6}=60^{\circ}\). A rotation of \(60^{\circ}\) (or any multiple of \(60^{\circ}\)) will map a regular hexagon onto itself.

Step2: Analyze reflection symmetry

A regular hexagon has reflection symmetry. If the line \(l\) is an axis of symmetry (a line that divides the hexagon into two congruent halves), then reflection across \(l\) will map the hexagon onto itself.

Step3: Check non - multiple of \(60^{\circ}\) rotations

For a rotation of \(45^{\circ}\), since \(45^{\circ}\) is not a multiple of \(60^{\circ}\) (\(\frac{360^{\circ}}{6} = 60^{\circ}\)), a rotation of \(45^{\circ}\) will not map the hexagon onto itself. For a rotation of \(72^{\circ}\), since \(72^{\circ}\) is not a multiple of \(60^{\circ}\) (\(\frac{360^{\circ}}{6}=60^{\circ}\)), a rotation of \(72^{\circ}\) will not map the hexagon onto itself.

Answer:

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