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.

(figure 1: a shape resembling a stylized 5)
rotational symmetry? ∘ yes ∘ no
angle? °

(figure 2: a star)
rotational symmetry? ∘ yes ∘ no
angle? °

Explanation:

Step1: Analyze the "5" - shaped figure

A figure has rotational symmetry if rotating it by a certain angle (less than \(360^\circ\)) makes it look the same. The "5" - shaped figure: when we try to rotate it, there's no angle (other than \(360^\circ\)) that will make it appear unchanged. So for the "5" - shaped figure:

• Rotational symmetry? No
• Angle? (Not applicable, but since it has no rotational symmetry, we note that)

Step2: Analyze the star (pentagram) figure

A regular pentagram (star - shaped figure with 5 points) has rotational symmetry. The formula for the smallest angle of rotational symmetry of a regular \(n\) - sided polygon (or a figure with \(n\) - fold rotational symmetry) is \(\frac{360^\circ}{n}\). For a pentagram, \(n = 5\). So the smallest angle of rotation is \(\frac{360^\circ}{5}=72^\circ\). So for the star:

• Rotational symmetry? Yes
• Angle? \(72^\circ\)

Answer:

For the "5" - shaped figure:

Rotational symmetry? No
Angle? (No valid angle as it has no rotational symmetry)

For the star - shaped figure:

Rotational symmetry? Yes
Angle? \(72^\circ\)