Question

the regular polygon below is to be rotated about its center. which angle of rotation would carry the figure? answer 180° 108° 225° 288°

Explanation:

Step1: Determine the number of sides.

The polygon is a regular decagon, so $n = 10$.

Step2: Use the rotation - angle formula.

The formula for the angle of rotation $\theta$ that maps a regular $n$-sided polygon onto itself is $\theta=\frac{360^{\circ}}{k}$, where $k$ is a positive integer and $k\leq n$. The minimum non - zero angle of rotation is $\frac{360^{\circ}}{n}$. For a regular polygon, angles of rotation that map the polygon onto itself are multiples of $\frac{360^{\circ}}{n}$. For a decagon ($n = 10$), $\frac{360^{\circ}}{n}=\frac{360^{\circ}}{10}=36^{\circ}$. We need to find which of the given angles is a multiple of $36^{\circ}$.

• For $180^{\circ}$, $\frac{180^{\circ}}{36^{\circ}} = 5$.
• For $108^{\circ}$, $\frac{108^{\circ}}{36^{\circ}}=3$.
• For $225^{\circ}$, $\frac{225^{\circ}}{36^{\circ}}=\frac{225}{36}=\frac{25}{4}$, not an integer.
• For $288^{\circ}$, $\frac{288^{\circ}}{36^{\circ}} = 8$.

Answer:

$180^{\circ},108^{\circ},288^{\circ}$