Question

which of the following transformations carry this regular polygon onto itself? rotation of 30° counterclockwise rotation of 90° counterclockwise rotation of 120° counterclockwise rotation of 36° counterclockwise

Explanation:

Step1: Calculate the rotation angle for a regular polygon

For a regular \(n -\)sided polygon, the angle of rotation that maps the polygon onto itself is \(\frac{360^{\circ}}{n}\).
Here, the polygon is a triangle (\(n = 3\)), so \(\frac{360^{\circ}}{3}=120^{\circ}\).

Step2: Generalize the rotation angles

Any multiple of \(\frac{360^{\circ}}{n}\) (i.e., \(k\times\frac{360^{\circ}}{n}\), where \(k = 1,2,\cdots,n\)) will also map the regular polygon onto itself. For \(n = 3\), when \(k = 1\), the angle is \(120^{\circ}\), when \(k = 2\), the angle is \(240^{\circ}\), and when \(k=3\), the angle is \(360^{\circ}\) (a full - rotation which is trivial).

Answer:

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