Question

1. alberto listed 90°, 180°, and 270° as the angles of rotation for which the figure has rotational symmetry. what was his error?

Explanation:

Rotational symmetry occurs when a figure can be rotated by an angle less than \(360^{\circ}\) and still look the same. For a regular octagon (the figure in the problem), the order of rotational symmetry is \(8\). The formula for the angle of rotational symmetry is \(\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For \(n = 8\), the angle is \(\frac{360^{\circ}}{8}=45^{\circ}\). So the angles of rotational symmetry should be multiples of \(45^{\circ}\) (\(45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ}\)). Alberto missed \(45^{\circ},135^{\circ},225^{\circ},315^{\circ}\).

Answer:

Alberto missed the angles \(45^{\circ},135^{\circ},225^{\circ},315^{\circ}\) which are also angles of rotational symmetry for the figure (a regular - like octagon). The correct angles of rotational symmetry are multiples of \(45^{\circ}\) (\(45^{\circ},90^{\circ},135^{\circ},180^{\circ},225^{\circ},270^{\circ},315^{\circ}\)).