Question

which of the following transformations carry this regular polygon onto itself?rotation of 36° clockwiserotation of 60° counterclockwiserotation of 72° counterclockwiserotation of 60° clockwise

Explanation:

Step1: Calculate the rotational symmetry angle

For a regular \(n -\)sided polygon, the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\). Here, \(n = 6\) (since it's a hexagon), so \(\frac{360^{\circ}}{6}=60^{\circ}\).

Step2: Analyze each rotation option

• A rotation of \(36^{\circ}\) clockwise: \(36^{\circ}\) is not a multiple of \(60^{\circ}\), so it won't map the hexagon onto itself.
• A rotation of \(60^{\circ}\) counter - clockwise: Since \(60^{\circ}\) is the rotational symmetry angle, this rotation will map the hexagon onto itself.
• A rotation of \(72^{\circ}\) counter - clockwise: \(72^{\circ}\) is not a multiple of \(60^{\circ}\), so it won't map the hexagon onto itself.
• A rotation of \(60^{\circ}\) clockwise: \(60^{\circ}\) is the rotational symmetry angle, so this rotation will map the hexagon onto itself.

Answer:

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