# 12.1: Exploring Symmetry

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Learn about lines of symmetry.
• Discuss line and rotational symmetry.
• Learn about the center of symmetry.

## Review Queue

1. Define symmetry in your own words.
2. Draw a regular hexagon. How many degrees does each angle have?
3. Draw all the diagonals in your hexagon. What is the measure of each central angle?
4. Plot the points \begin{align*}A(1, 3), B(3, 1), C(5, 3)\end{align*}, and \begin{align*}D(3, 5)\end{align*}. What kind of shape is this? Prove it using the distance formula and/or slope.

Know What? Symmetry exists all over nature. One example is a starfish, like the one below. Draw in the line(s) of symmetry, center of symmetry and the angle of rotation for this starfish.

## Lines of Symmetry

Line of Symmetry: A line that passes through a figure such that it splits the figure into two congruent halves.

Many figures have a line of symmetry, but some do not have any lines of symmetry. Figures can also have more than one line of symmetry.

Example 1: Find all lines of symmetry for the shapes below.

a)

b)

c)

d)

Solution: For each figure, draw lines that cut the figure in half perfectly. Figure a) has two lines of symmetry, b) has eight, c) has no lines of symmetry, and d) has one.

a)

b)

c)

d)

Figures a), b), and d) all have line symmetry.

Line Symmetry: When a figure has one or more lines of symmetry.

Example 2: Do the figures below have line symmetry?

a)

b)

Solution: Yes, both of these figures have line symmetry. One line of symmetry is shown for the flower; however it has several more lines of symmetry. The butterfly only has one line of symmetry.

## Rotational Symmetry

Rotational Symmetry: When a figure can be rotated (less that \begin{align*}360^\circ\end{align*}) and it looks the same way it did before the rotation.

Center of Rotation: The point at which the figure is rotated around such that the rotational symmetry holds. Typically, the center of rotation is the center of the figure.

Along with rotational symmetry and a center of rotation, figures will have an angle of rotation. The angle of rotation, tells us how many degrees we can rotate a figure so that it still looks the same.

Example 3: Determine if each figure below has rotational symmetry. If it does, determine the angle of rotation.

a)

b)

c)

Solution:

a) The regular pentagon can be rotated 5 times so that each vertex is at the top. This means the angle of rotation is \begin{align*}\frac{360^\circ}{5} = 72^\circ\end{align*}.

The pentagon can be rotated \begin{align*}72^\circ, 144^\circ, 216^\circ\end{align*}, and \begin{align*}288^\circ\end{align*} so that it still looks the same.

b) The \begin{align*}“N”\end{align*} can be rotated twice, \begin{align*}180^\circ\end{align*}, so that it still looks the same.

c) The checkerboard can be rotated 4 times so that the angle of rotation is \begin{align*}\frac{360^\circ}{4} = 90^\circ\end{align*}. It can be rotated \begin{align*}180^\circ\end{align*} and \begin{align*}270^\circ\end{align*} as well. The final rotation is always \begin{align*}360^\circ\end{align*} to get the figure back to its original position.

In general, if a shape can be rotated n times, the angle of rotation is \begin{align*}\frac{360^\circ}{n}\end{align*}. Then, multiply the angle of rotation by 1, 2, 3..., and \begin{align*}n\end{align*} to find the additional angles of rotation.

Know What? Revisited The starfish has 5 lines of symmetry and has rotational symmetry of \begin{align*}72^\circ\end{align*}. Therefore, the starfish can be rotated \begin{align*}72^\circ, 144^\circ, 216^\circ\end{align*}, and \begin{align*}288^\circ\end{align*} and it will still look the same. The center of rotation is the center of the starfish.

## Review Questions

Determine if the following questions are ALWAYS true, SOMETIMES true, or NEVER true.

1. Right triangles have line symmetry.
2. Isosceles triangles have line symmetry.
3. Every rectangle has line symmetry.
4. Every rectangle has exactly two lines of symmetry.
5. Every parallelogram has line symmetry.
6. Every square has exactly two lines of symmetry.
7. Every regular polygon has three lines of symmetry.
8. Every sector of a circle has a line of symmetry.
9. Every parallelogram has rotational symmetry.
10. A rectangle has \begin{align*}90^\circ, 180^\circ\end{align*}, and \begin{align*}270^\circ\end{align*} angles of rotation.
11. Draw a quadrilateral that has two pairs of congruent sides and exactly one line of symmetry.
12. Draw a figure with infinitely many lines of symmetry.
13. Draw a figure that has one line of symmetry and no rotational symmetry.
14. Fill in the blank: A regular polygon with \begin{align*}n\end{align*} sides has ______ lines of symmetry.

Find all lines of symmetry for the letters below.

1. Do any of the letters above have rotational symmetry? If so, which one(s) and what are the angle(s) of rotation?

Determine if the words below have line symmetry or rotational symmetry.

1. OHIO
2. MOW
3. WOW
4. KICK
5. pod

Trace each figure and then draw in all lines of symmetry.

Find the angle(s) of rotation for each figure below.

Determine if the figures below have line symmetry or rotational symmetry. Identify all lines of symmetry and all angles of rotation.

1. Where one side of an object matches the other side; answers will vary.

2 and 3. each angle has \begin{align*}\frac{(n-1)180^\circ}{n} = \frac{5(180^\circ)}{6} = 120^\circ\end{align*}

each central angle has \begin{align*}\frac{360^\circ}{6} = 60^\circ\end{align*}

4. The figure is a square.

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