Question

analyzing rotational symmetry
if this regular decagon is rotated counterclockwise by
3 times the smallest angle of rotation, which vertex
will be in the top position?

Explanation:

Step1: Calculate the smallest rotation angle

The formula for the smallest rotation angle of a regular \(n -\)gon is \(\frac{360^{\circ}}{n}\). For a decagon (\(n = 10\)), the smallest rotation angle \(\theta=\frac{360^{\circ}}{10}=36^{\circ}\).

Step2: Calculate the total rotation angle

The total rotation angle is \(3\times\theta\). Substitute \(\theta = 36^{\circ}\), we get \(3\times36^{\circ}=108^{\circ}\).

Step3: Determine the vertex after rotation

Since each vertex is separated by \(36^{\circ}\), a \(108^{\circ}\) rotation (counter - clockwise) means moving \( \frac{108^{\circ}}{36^{\circ}}=3\) vertices counter - clockwise from the initial vertex \(B\). Moving 3 vertices counter - clockwise from \(B\) ( \(B
ightarrow C\) (1st move), \(C
ightarrow D\) (2nd move), \(D
ightarrow E\) (3rd move)).

Answer:

\(E\)