Question

a regular hexagon is rotated ( 360^circ ) about its center. how many times does the image of the hexagon coincide with the preimage during the rotation?
a. 1 time
b. 6 times
c. 12 times
d. 3 times

Explanation:

Step1: Recall properties of regular hexagon

A regular hexagon has 6 equal sides and 6 equal angles. The central angle between two adjacent vertices from the center is $\frac{360^\circ}{6} = 60^\circ$.

Step2: Determine coincidence during rotation

When rotating a regular hexagon about its center, the image coincides with the pre - image every time it rotates by a multiple of the central angle ($60^\circ$). To rotate a full $360^\circ$, we calculate the number of times it coincides by dividing the total rotation angle ($360^\circ$) by the angle of each coincidence ($60^\circ$). So, $\frac{360^\circ}{60^\circ}=6$. But we also count the starting position, so the number of times the image coincides with the pre - image during a $360^\circ$ rotation is 6 (including the start and end, but the question is about how many times during the rotation, so from $0^\circ$ to $360^\circ$, excluding the final $360^\circ$ which is the same as the start, the number of coincidences is 6? Wait, no. Wait, when you start at $0^\circ$, that's the first position. Then after rotating $60^\circ$, it coincides again, then $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and then $360^\circ$ is the same as $0^\circ$. So during the rotation (from $0^\circ$ up to but not including $360^\circ$), the number of times it coincides is 6? Wait, no, the question says "during the rotation". Let's think again. A regular hexagon has rotational symmetry of order 6. The order of rotational symmetry is the number of times a figure coincides with itself as it rotates through $360^\circ$. So when rotating $360^\circ$ about its center, the image coincides with the pre - image 6 times (at $0^\circ$, $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$). But wait, the starting position is the pre - image. So if we consider the rotation from $0^\circ$ to $360^\circ$, the number of times the image coincides with the pre - image (excluding the final $360^\circ$ which is the same as the start) is 6? Wait, no, the order of rotational symmetry is 6, which means that as it rotates $360^\circ$, it coincides with itself 6 times (including the initial position). So the answer should be 6 times.

Answer:

B. 6 times