Question

which of the following transformations carry this regular polygon onto itself?
rotation of 60° counterclockwise
rotation of 120° counterclockwise
rotation of 72° counterclockwise
reflection across l

Explanation:

Step1: Calculate the rotation angle of a regular pentagon

The formula for the central angle of a regular \(n -\)sided polygon is \(\theta=\frac{360^{\circ}}{n}\). For a regular pentagon, \(n = 5\), so \(\theta=\frac{360^{\circ}}{5}=72^{\circ}\). A rotation of \(k\times72^{\circ}(k = 1,2,3,4,5)\) will map the regular pentagon onto itself. Rotations of \(60^{\circ}\) and \(120^{\circ}\) are not multiples of \(72^{\circ}\), so they do not map the pentagon onto itself.

Step2: Analyze the reflection

A reflection across a line of symmetry (in a regular pentagon, there are 5 lines of symmetry) will map the polygon onto itself. If line \(l\) is a line of symmetry (for example, a line passing through a vertex and the mid - point of the opposite side in a regular pentagon), then reflection across line \(l\) will map the pentagon onto itself.

Answer:

rotation of \(72^{\circ}\) counterclockwise, reflection across \(l\)