Question

applications

1. because of the beauty of symmetry, designs are often made that use regular polygons to take advantage of this symmetry. a common shape based on a square is shown below.

(a) draw all lines of symmetry for this design.
(b) what is the minimum angle of rotation needed to map this figure onto itself?
(c) how do the symmetries of this figure compare to those of a simple square?

Explanation:

(a)

A line of symmetry is a line that divides a figure into two congruent parts. For a square - based figure with the given design:

• Vertical line of symmetry: A vertical line passing through the center of the square (and the figure).
• Horizontal line of symmetry: A horizontal line passing through the center of the square (and the figure).
• Two diagonal lines of symmetry: The two lines that connect the opposite vertices of the square (and pass through the center of the figure).

(b)

The formula for the minimum angle of rotation \(\theta\) of a regular \(n -\)sided polygon (or a figure with \(n\) - fold rotational symmetry) is \(\theta=\frac{360^{\circ}}{n}\).
Since the figure has \(4\) - fold rotational symmetry (similar to a square), \(n = 4\).

$$ \theta=\frac{360^{\circ}}{4}=90^{\circ} $$

(c)

• Line of symmetry comparison:
• A simple square has \(4\) lines of symmetry (vertical, horizontal, and two diagonals). The given figure also has \(4\) lines of symmetry (same as the square).
• Rotational symmetry comparison:
• A simple square has a minimum angle of rotation of \(90^{\circ}\) (because \(n = 4\) in \(\theta=\frac{360^{\circ}}{n}\)). The given figure also has a minimum angle of rotation of \(90^{\circ}\) (since it has \(4\) - fold rotational symmetry).

Answer:

(a) Draw 4 lines of symmetry: vertical, horizontal, and two diagonals.
(b) \(90^{\circ}\)
(c) The number of lines of symmetry and the minimum angle of rotation are the same as those of a simple square.