Question

the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure onto itself
answer
144 324
225 90

Explanation:

Step1: Determine the number of sides of the polygon

The given polygon is a decagon (10 - sided polygon).

Step2: Use the formula for the angle of rotation of a regular polygon

The formula for the angle of rotation of a regular polygon is \(\frac{360^{\circ}}{n}\), where \(n\) is the number of sides. For a decagon (\(n = 10\)), the smallest angle of rotation is \(\frac{360^{\circ}}{10}=36^{\circ}\). Any multiple of \(36^{\circ}\) will also carry the polygon onto itself.

Step3: Check each option

• For \(144^{\circ}\): \(\frac{144^{\circ}}{36^{\circ}} = 4\) (a whole number).
• For \(225^{\circ}\): \(\frac{225^{\circ}}{36^{\circ}}=\frac{225}{36}=\frac{25}{4}\) (not a whole number).
• For \(324^{\circ}\): \(\frac{324^{\circ}}{36^{\circ}} = 9\) (a whole number).
• For \(90^{\circ}\): \(\frac{90^{\circ}}{36^{\circ}}=\frac{5}{2}\) (not a whole number).

Answer:

\(144^{\circ}\) and \(324^{\circ}\)