Question

select the correct answer from each drop - down menu.
abcdefgh is a regular octagon. the minimum degree of rotation by which this octagon can map onto itself is
, it will take
increments of that degree of clockwise rotation for a (the image of a) to coincide with c.

Explanation:

Step1: Find the central angle of a regular octagon

A regular octagon has 8 equal sides, so the central angle between two adjacent vertices is calculated by dividing the total angle around a point ($360^\circ$) by the number of sides ($n = 8$). The formula for the central angle $\theta$ is $\theta=\frac{360^\circ}{n}$. Substituting $n = 8$, we get $\theta=\frac{360^\circ}{8}=45^\circ$.

Step2: Determine the rotation from A to C

To go from vertex A to vertex C in a regular octagon, we move 2 edges (from A to B to C). So the angle of rotation is $2\times45^\circ = 90^\circ$.

Step3: Check the number of increments (if needed, but here we found the rotation angle)

The minimum degree of rotation for the octagon to map onto itself is the central angle, which is $45^\circ$, and the rotation from A to C is $90^\circ$ (2 increments of $45^\circ$ each).

Answer:

The minimum degree of rotation for the octagon to map onto itself is $45^\circ$, and the rotation from A to C is $90^\circ$ (so the number of increments is 2, but the main angle for self - mapping is $45^\circ$ and from A to C is $90^\circ$). If we consider the first drop - down (minimum rotation for self - mapping): $45^\circ$, and the second (rotation from A to C): $90^\circ$ (or 2 increments of $45^\circ$).