Question

the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure onto itself? answer 315° 150° 225° 144°

Explanation:

Step1: Recall rotation - symmetry formula

For a regular \(n\) - sided polygon, the angle of rotation \(\theta\) that maps the polygon onto itself is given by \(\theta=\frac{360^{\circ}}{k}\), where \(k = 1,2,\cdots,n\). Here, the polygon is a pentagon, so \(n = 5\). The basic angle of rotation is \(\frac{360^{\circ}}{5}=72^{\circ}\), and other angles are multiples of \(72^{\circ}\), i.e., \(\theta = 72^{\circ}\times k,k\in\mathbb{Z}\).

Step2: Check the options

We check each option to see if it is a multiple of \(72^{\circ}\).

• For \(315^{\circ}\), \(\frac{315}{72}=\frac{35}{8}\), not an integer.
• For \(150^{\circ}\), \(\frac{150}{72}=\frac{25}{12}\), not an integer.
• For \(225^{\circ}\), \(\frac{225}{72}=\frac{25}{8}\), not an integer.
• For \(144^{\circ}\), \(\frac{144}{72} = 2\), an integer.

Answer:

\(144^{\circ}\)