Question

which of the following transformations carry this regular polygon onto itself?
rotation of 45° counterclockwise
rotation of 60° counterclockwise
reflection across l
rotation of 90° counterclockwise

Explanation:

Step1: Recall rotational symmetry formula

For a regular polygon with \(n\) sides, the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\). A square (\(n = 4\)) has rotational symmetry angles of \(\frac{360^{\circ}}{4}=90^{\circ},180^{\circ},270^{\circ},360^{\circ}\). So a rotation of \(90^{\circ}\) counter - clockwise will map the square onto itself.

Step2: Analyze reflection

A reflection across a line of symmetry will map a figure onto itself. For a square, if line \(l\) is a line of symmetry (e.g., a diagonal or a perpendicular bisector of a side), reflection across \(l\) will map the square onto itself.

Answer:

reflection across \(l\), rotation of \(90^{\circ}\) counterclockwise