Question

which of the following transformations carry this regular polygon onto itself? rotation of 40° counterclockwise reflection across l rotation of 45° counterclockwise rotation of 40° 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, the polygon is an octagon (\(n = 8\)), so \(\frac{360^{\circ}}{8}=45^{\circ}\). A rotation of \(40^{\circ}\) (clock - wise or counter - clockwise) will not map the octagon onto itself because \(40\) is not a divisor of \(360\) for an octagon.

Step2: Analyze reflection

A reflection across a line of symmetry (such as the line \(l\) which is a line of symmetry for the octagon) will map the regular polygon onto itself.

Answer:

reflection across \(l\), rotation of \(45^{\circ}\) counterclockwise