Question

which of the following transformations carry this regular polygon onto itself? rotation of 72° counterclockwise reflection across l₂ rotation of 60° counterclockwise reflection across l₁

Explanation:

Step1: Recall rotation property of regular hexagon

A regular hexagon has order of rotational symmetry 6. The central angle of a regular hexagon is $\frac{360^{\circ}}{6}=60^{\circ}$. A rotation of $60^{\circ}$ and its multiples will map the hexagon onto itself. A $72^{\circ}$ rotation will not map it onto itself as $360\div72 = 5$ and the hexagon's symmetry is based on 6 - fold rotational symmetry.

Step2: Recall reflection property

The line $l_2$ is a line of symmetry for the regular hexagon as it passes through opposite vertices. Reflection across a line of symmetry maps a figure onto itself. The line $l_1$ is not a line of symmetry for the regular hexagon. A $60^{\circ}$ counter - clockwise rotation will map the regular hexagon onto itself since $60^{\circ}$ is a divisor of $360^{\circ}$ and is related to the hexagon's rotational symmetry.

Answer:

B. reflection across $l_2$, C. rotation of $60^{\circ}$ counterclockwise