Question

a nonagon is a nine - sided polygon. if a regular nonagon was rotated about its center point, which of the following angles would map the figure onto itself?
80°
40°
120°
60°

Explanation:

Step1: Recall rotation formula for regular polygon

The angle of rotation $\theta$ that maps a regular $n$-sided polygon onto itself is given by $\theta=\frac{360^{\circ}}{k}$, where $k$ is a positive - integer and $n$ is the number of sides of the polygon. For a non - agon, $n = 9$. The minimum non - zero angle of rotation that maps a regular non - agon onto itself is when $k = 1$, and $\theta=\frac{360^{\circ}}{9}=40^{\circ}$. Also, multiples of this angle will map the non - agon onto itself.

Step2: Check the given options

We check each option:

• Option 80°: $\frac{80^{\circ}}{40^{\circ}} = 2$, so a rotation of 80° is a valid rotation that maps the non - agon onto itself.
• Option 40°: As calculated above, 40° is the minimum non - zero rotation angle that maps the non - agon onto itself.
• Option 120°: $\frac{120^{\circ}}{40^{\circ}}=3$, so a rotation of 120° is a valid rotation that maps the non - agon onto itself.
• Option 60°: $\frac{60^{\circ}}{40^{\circ}} = 1.5$, so 60° is not a valid rotation angle that maps the non - agon onto itself.

Answer:

A. 80°, B. 40°, C. 120°