Question

which of the following transformations carry this regular polygon onto itself?
rotation of 60° counterclockwise
reflection across l
rotation of 60° clockwise
rotation of 36° counterclockwise

Explanation:

Step1: Rotational Symmetry of Regular Hexagon

A regular hexagon has rotational symmetry of order \(6\). The formula for the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\), where \(n = 6\) (number of sides). So \(\frac{360^{\circ}}{6}=60^{\circ}\). A rotation of \(60^{\circ}\) (clock - wise or counter - clockwise) will map the hexagon onto itself.

Step2: Reflection Symmetry

A regular hexagon has \(6\) lines of symmetry. If line \(l\) is a line of symmetry (as it appears to bisect opposite sides in the given figure), a reflection across line \(l\) will map the hexagon onto itself.

Step3: Rotation of \(36^{\circ}\)

Since \(\frac{360^{\circ}}{36^{\circ}} = 10
eq6\), a rotation of \(36^{\circ}\) will not map the hexagon onto itself.

Answer:

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