Question

for each figure below, determine if it has rotational symmetry. if it does, give the smallest angle of rotation needed for the figure to appear unmoved. rotational symmetry? ○ yes ○ no angle? □° rotational symmetry? ○ yes ○ no angle? □°

Explanation:

Left Figure (Star - like with 5 arms)

Step1: Count the number of identical parts

The left figure has 5 identical arms. For rotational symmetry, the smallest angle of rotation ($\theta$) is given by $\theta=\frac{360^\circ}{n}$, where $n$ is the number of identical parts. Here, $n = 5$.

Step2: Calculate the angle

$\theta=\frac{360^\circ}{5}=72^\circ$. So it has rotational symmetry.

Step1: Count the number of sides (identical parts)

The right figure is a regular octagon, so $n = 8$ (number of identical sides/parts).

Step2: Calculate the angle

Using the formula $\theta=\frac{360^\circ}{n}$, with $n = 8$, we get $\theta=\frac{360^\circ}{8}=45^\circ$. So it has rotational symmetry.

Answer:

Rotational symmetry? Yes
Angle? $72^\circ$

Right Figure (Octagon - 8 - sided polygon)