Question

1. a regular polygon has rotational symmetry with an order of 5 and a magnitude of 72°. what is the figure?

Explanation:

Step1: Recall rotational symmetry of regular polygons

For a regular polygon, the order of rotational symmetry is equal to the number of sides \( n \). The magnitude of rotational symmetry (the smallest angle of rotation that maps the polygon onto itself) is given by \( \frac{360^{\circ}}{n} \).

Step2: Use the formula to find the number of sides

We know the order of rotational symmetry (which is equal to the number of sides \( n \)) and the magnitude of rotation. The formula for the magnitude of rotational symmetry \( \theta \) is \( \theta=\frac{360^{\circ}}{n} \). We are given that the order of rotational symmetry is 5, so \( n = 5 \), and we can also verify using the magnitude. Given \( \theta=72^{\circ} \), from \( \theta=\frac{360^{\circ}}{n} \), we can solve for \( n \): \( n=\frac{360^{\circ}}{\theta} \). Substituting \( \theta = 72^{\circ} \), we get \( n=\frac{360^{\circ}}{72^{\circ}} = 5 \). A regular polygon with 5 sides is a regular pentagon.

Answer:

Regular Pentagon