Question

which of the following transformations carry this regular polygon onto itself?rotation of 72° counterclockwiserotation of 40° counterclockwiserotation of 36° clockwiserotation of 60° clockwise

Explanation:

Step1: Calculate the rotational symmetry angle

For a regular polygon with \(n\) sides, the rotational symmetry angle \(\theta=\frac{360^{\circ}}{n}\). A regular hexagon has \(n = 6\) sides. So \(\theta=\frac{360^{\circ}}{6}=60^{\circ}\).

Step2: Check each option

• For a rotation of \(72^{\circ}\): \(\frac{360^{\circ}}{72^{\circ}} = 5\), not relevant to a hexagon.
• For a rotation of \(40^{\circ}\): \(\frac{360^{\circ}}{40^{\circ}}=9\), not relevant to a hexagon.
• For a rotation of \(36^{\circ}\): \(\frac{360^{\circ}}{36^{\circ}} = 10\), not relevant to a hexagon.
• For a rotation of \(60^{\circ}\): Since the rotational symmetry angle of a hexagon is \(60^{\circ}\), a rotation of \(60^{\circ}\) clockwise (or counter - clockwise) will map the hexagon onto itself.

Answer:

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