Question

which of the following transformations carry this regular polygon onto itself?rotation of 90° counterclockwiserotation of 120° counterclockwiserotation of 120° clockwiserotation of 30° clockwise

Explanation:

Step1: Recall the formula for rotational symmetry of a regular polygon

The formula for the angle of rotational symmetry of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a regular triangle (\(n = 3\)), the angle of rotational symmetry is \(\frac{360^{\circ}}{3}=120^{\circ}\).

Step2: Analyze each rotation option

• For a rotation of \(90^{\circ}\) counter - clockwise: Since \(90^{\circ}\) is not a multiple of \(120^{\circ}\), a rotation of \(90^{\circ}\) will not map the regular triangle onto itself.
• For a rotation of \(120^{\circ}\) counter - clockwise: Since \(120^{\circ}\) is the angle of rotational symmetry of a regular triangle, a rotation of \(120^{\circ}\) counter - clockwise will map the regular triangle onto itself.
• For a rotation of \(120^{\circ}\) clockwise: Since \(120^{\circ}\) is the angle of rotational symmetry of a regular triangle, a rotation of \(120^{\circ}\) clockwise will map the regular triangle onto itself.
• For a rotation of \(30^{\circ}\) clockwise: Since \(30^{\circ}\) is not a multiple of \(120^{\circ}\), a rotation of \(30^{\circ}\) will not map the regular triangle onto itself.

Answer:

rotation of \(120^{\circ}\) counterclockwise, rotation of \(120^{\circ}\) clockwise