Question

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

Explanation:

Step1: Find rotation symmetry angle

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

Step2: Count coincidences

As we rotate from $0^{\circ}$ to $360^{\circ}$, the hexagon coincides with itself when rotated by multiples of $60^{\circ}$: at $60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}, 360^{\circ}$. Including the start (0°), but during rotation (excluding final 360° as same as start), we have 6 coincidences (including the initial position? Wait, when rotating 360°, the number of times it coincides: the rotation symmetry order is 6. So during the rotation (from 0 to 360, not including 360 as same as start), the number of times it coincides is 6 (at 60, 120, 180, 240, 300, and 360 which is the end, but the question says "during the rotation"—so from 0 to 360, the number of times the image coincides with preimage: the order of rotational symmetry is 6, so it coincides 6 times (including the start? Wait, when you start rotating, at 0° it's the same, then at 60°, 120°,..., 360° (which is 0° again). So during the rotation (as it rotates 360°), the number of times is 6 (since 360/60 = 6, so at each 60° increment, including 0° and 360°, but 0° is the start, 360° is the end. But the question is "during the rotation"—so how many times does the image coincide with preimage as it's rotated 360°? So from 0 to 360, the angles where it coincides are 0°, 60°, 120°, 180°, 240°, 300°, 360°—but 0° and 360° are the same. Wait, no: the rotational symmetry order is the number of times it maps onto itself when rotated 360°, which is 6. So during the rotation (as it goes from 0 to 360), it coincides 6 times (at 60°, 120°, 180°, 240°, 300°, and 360°—but 360° is the end, same as start. But the answer options: option C is 6 times. So the correct answer is 6.

Answer:

C. 6 times