Question

identify all the lines of symmetry for the figure on the right. if the figure has rotational symmetry, tell the angle of rotation. which choice below shows all the lines of symmetry for the figure?
a. image of a star - like figure with a vertical line of symmetry

b. image of a star - like figure with a horizontal line of symmetry

c. image of a star - like figure with multiple lines of symmetry

d. the figure has no lines of symmetry.

Explanation:

Step1: Analyze the figure's symmetry

The given figure is a star - like shape. A line of symmetry is a line that divides a figure into two congruent halves that are mirror images of each other. For a regular star - shaped figure (like a regular hexagram or other symmetric star figures), we need to check the options.
Looking at the options, option C shows multiple lines of symmetry that pass through the vertices and the mid - points of the opposite sides (or in the case of a star, through the points of the star and the opposite "valleys"). The figure has rotational symmetry as well. To find the angle of rotational symmetry, we know that for a figure with \(n\) lines of symmetry (in the case of a regular star - like figure with 6 - fold symmetry, \(n = 6\)), the angle of rotation \(\theta=\frac{360^{\circ}}{n}\). Here, if we assume the figure has 6 - fold symmetry (from the appearance of the star with 6 points), the angle of rotation is \(\frac{360^{\circ}}{6}=60^{\circ}\). But first, for the lines of symmetry, option C correctly represents all the lines of symmetry as it shows the lines passing through each of the star's points and the opposite sides (or equivalent symmetric axes).

Step2: Determine the correct option for lines of symmetry

By visually inspecting the figure and the options, option C shows all the lines of symmetry. The figure is symmetric with respect to multiple lines that pass through the center and the vertices (or the equivalent symmetric points) of the star - shaped figure. Option A and B show only one or two lines, which is incorrect for this symmetric star - like figure. Option D is incorrect as the figure clearly has lines of symmetry.

Answer:

C. (The option that shows all the lines of symmetry for the star - like figure) and the angle of rotational symmetry is \(60^{\circ}\) (since \(\frac{360^{\circ}}{6} = 60^{\circ}\) assuming 6 - fold symmetry from the figure's appearance). But for the multiple - choice part about lines of symmetry, the answer is C.