Question

part c
how can you use the measures of arcs created by an inscribed regular polygon to determine the rotations that will map the polygon onto itself?
show hints
divide 90 by the number of arcs.
divide 270 by the number of arcs.
divide 360 by the number of arcs.
divide 180 by the number of arcs.

Explanation:

Step1: Recall rotational symmetry concept

For a regular polygon, the angles of rotation that map the polygon onto itself are given by $\frac{360^{\circ}}{n}$, where $n$ is the number of sides (and also the number of arcs created by the inscribed polygon). If we know the measures of arcs, we can find the angles of rotation by dividing the common - multiples of 360 (like 90, 180, 270, 360) by the number of arcs.
Let the number of arcs be $n$. The angles of rotation $\theta$ that map the polygon onto itself are such that $\theta=\frac{k\cdot360^{\circ}}{n}$, where $k = 1,2,\cdots,n$. When we divide 90, 180, 270, 360 by $n$, we get the possible non - zero angles of rotation less than or equal to 360 degrees.
For example, if we divide 360 by $n$, we get the smallest non - zero angle of rotation for a regular polygon (the angle between consecutive vertices when rotating about the center). If we divide 180 by $n$, we get an angle of rotation that may map the polygon onto itself depending on the value of $n$. If $n$ is even, dividing 180 by $n$ will give a valid angle of rotation. Similarly for 270 and 90.

Answer:

The angles of rotation that map a regular polygon onto itself can be found by dividing 90, 180, 270, 360 by the number of arcs created by the inscribed regular polygon. These values represent possible non - zero angles of rotation (less than or equal to 360 degrees) about the center of the polygon that will result in the polygon coinciding with its original position.