Question

1. a. what are the rotational symmetries for the figure? does the figure have point symmetry?

Explanation:

Step1: Determine rotational symmetry angles

Rotational symmetry occurs when a figure can be rotated by an angle \(\theta\) (\(0^{\circ}<\theta\leqslant360^{\circ}\)) and still look the same. For a four - pointed star, the formula for the angle of rotational symmetry is \(\theta=\frac{360^{\circ}}{n}\), where \(n\) is the number of congruent parts. Here \(n = 4\). So \(\theta=\frac{360^{\circ}}{4}=90^{\circ}\). The rotational symmetries are \(90^{\circ},180^{\circ},270^{\circ}\) (since \(90^{\circ}\times1 = 90^{\circ}\), \(90^{\circ}\times2=180^{\circ}\), \(90^{\circ}\times3 = 270^{\circ}\))

Step2: Check for point symmetry

Point symmetry (also known as 180 - degree rotational symmetry) means that for every point \(P\) on the figure, there is a point \(P'\) such that the center of the figure is the mid - point of the segment \(PP'\). A rotation of \(180^{\circ}\) maps the four - pointed star onto itself.

Answer:

The rotational symmetries are \(90^{\circ},180^{\circ},270^{\circ}\). The figure has point symmetry (because of the \(180^{\circ}\) rotational symmetry).