Question

select the correct answer.
a regular octagon rotates 360° about its center. how many times does the image of the octagon coincide with the preimage during the rotation?
a. 1
b. 2
c. 4
d. 8

Explanation:

Step1: Understand rotational symmetry

A regular octagon has 8 equal sides and angles. The rotational symmetry of a regular polygon is determined by its number of sides. The angle of rotation for which the polygon coincides with itself is $\frac{360^\circ}{n}$, where $n$ is the number of sides.

Step2: Calculate coincidence times

For a regular octagon, $n = 8$. The angle of rotation for coincidence is $\frac{360^\circ}{8}= 45^\circ$. As it rotates $360^\circ$, we find how many times it coincides: at $0^\circ, 45^\circ, 90^\circ, 135^\circ, 180^\circ, 225^\circ, 270^\circ, 315^\circ, 360^\circ$. But we count the number of times during the rotation (including start and end? Wait, the question is "during the rotation". Wait, when rotating 360°, the number of times the image coincides with the preimage: a regular octagon, each time it rotates by a multiple of 45°, it coincides. So from 0° (start) to 360° (end), the number of times is 8? Wait, no: when you rotate 360°, the number of distinct positions where it coincides: for a regular $n$-gon, the number of times it coincides with itself when rotating 360° about its center is $n$. Because at each $\frac{360^\circ}{n}$ rotation, it coincides. So for $n = 8$, it's 8 times? Wait, but let's think again. Wait, the start position (0°) and then after each 45° rotation: 45°, 90°, ..., 360° (which is same as 0°). So how many times does it coincide during the rotation? Let's list the angles: 0° (start), 45°, 90°, 135°, 180°, 225°, 270°, 315°, 360° (end). But is 360° considered "during" the rotation? The question says "rotates 360° about its center. How many times does the image... coincide with the preimage during the rotation?" So when rotating from 0° to 360°, the number of times it coincides: at 0° (initial), then at 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360° (which is same as 0°). But maybe the question counts the number of times including the start and end, but for a regular octagon, the number of rotational symmetries (including identity) is 8. So the answer should be 8? Wait, but let's check the options. Option D is 8. Wait, maybe I made a mistake earlier. Wait, a regular octagon, when rotating 360°, the number of times it coincides with itself: each 45° rotation, so 360/45 = 8 times. So the answer is 8.

Answer:

D. 8