Question

the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure onto itself? answer 270° 324° 240° 252°

Explanation:

Step1: Recall rotation - symmetry formula

For a regular polygon with \(n\) sides, the angle of rotation \(\theta\) that maps the polygon onto itself is given by \(\theta=\frac{360^{\circ}}{k}\), where \(k = 1,2,\cdots,n\). A regular hexagon has \(n = 6\) sides. The possible angles of rotation are \(\frac{360^{\circ}}{1}=360^{\circ}\), \(\frac{360^{\circ}}{2} = 180^{\circ}\), \(\frac{360^{\circ}}{3}=120^{\circ}\), \(\frac{360^{\circ}}{4} = 90^{\circ}\), \(\frac{360^{\circ}}{5}=72^{\circ}\), \(\frac{360^{\circ}}{6}=60^{\circ}\), and their multiples.

Step2: Check each option

• For \(270^{\circ}\), \(\frac{360}{270}=\frac{4}{3}\), not an integer.
• For \(240^{\circ}\), \(\frac{360}{240}=\frac{3}{2}\), not an integer.
• For \(324^{\circ}\), \(\frac{360}{324}=\frac{10}{9}\), not an integer.
• For \(252^{\circ}\), \(\frac{360}{252}=\frac{10}{7}\), not an integer. But \(240^{\circ}=4\times60^{\circ}\), and \(60^{\circ}\) is a valid angle of rotation for a regular hexagon.

Answer:

240°