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# Rotation Symmetry

## When a figure can be rotated less than 360 degrees and look the same as it did before.

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Rotation Symmetry
What if you were asked to consider the presence of symmetry in nature? The starfish, below, is one example of symmetry in nature. Draw in the center of symmetry and the angle of rotation for this starfish.

### Rotation Symmetry

A figure exhibits rotational symmetry if it can be rotated (less than 360\begin{align*}360^\circ\end{align*}) and look the same as it did before the rotation. The center of rotation is the point that 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 is the angle of rotation that tells us how many degrees to rotate the figure so that it still looks the same. In general, if a shape can be rotated n times, the angle of rotation is 360n.\begin{align*}\frac{360^\circ}{n}.\end{align*} Multiply the angle of rotation by 1, 2, 3...n\begin{align*}n\end{align*} to find the additional angles of rotation. The number of rotations may be referred to as the order.

#### Recognizing Rotational Symmetry

1. Determine if the figure below has rotational symmetry. Find the angle of rotational symmetry and the order.

A regular pentagon can be rotated 5 times to demonstrate rotational symmetry, so it is order 5. Because there are 5 lines of rotational symmetry, the angle would be 3605=72.\begin{align*}\frac{360^\circ}{5}= 72^\circ.\end{align*}

2. Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.

The N\begin{align*}N\end{align*} can be rotated twice, so it is order 2. The angle of rotation is 180.\begin{align*}180^\circ.\end{align*}

3. Determine if the figure below has rotational symmetry. Find the angle and how many times it can be rotated.

The checkerboard can be rotated 4 times. There are 4 lines of rotational symmetry, so the angle of rotation is 3604=90.\begin{align*}\frac{360^\circ}{4}=90^\circ.\end{align*} It appears the same at 0(or 360),90,180, and 270.\begin{align*}0^\circ (\text{or 360}^\circ), 90^\circ, 180^\circ, \text{ and }270^\circ.\end{align*}

#### Starfish Problem Revisited

The starfish has rotational symmetry of 72\begin{align*}72^\circ\end{align*}. Therefore, the starfish can be rotated 72,144,216,288,\begin{align*}72^\circ, 144^\circ, 216^\circ, 288^\circ,\end{align*} and 360\begin{align*}360^\circ\end{align*} and it will still look the same. The center of rotation is the center of the starfish.

### Examples

Find the angle of rotation and the number of times each figure can rotate.

#### Example 1

The parallelogram can be rotated twice, so it is order 2. The angle of rotation is 180.\begin{align*}180^\circ.\end{align*}

#### Example 2

The hexagon can be rotated six times, order 6. The angle of rotation is 60.\begin{align*}60^\circ.\end{align*}

#### Example 3

This figure can be rotated four times, order 4. The angle of rotation is 90.\begin{align*}90^\circ.\end{align*}

### Review

1. If a figure has 3 lines of rotational symmetry, it can be rotated _______ times.
2. If a figure can be rotated 6 times, it has _______ lines of rotational symmetry.
3. If a figure can be rotated n\begin{align*}n\end{align*} times, it has _______ lines of rotational symmetry.
4. To find the angle of rotation, divide 360\begin{align*}360^\circ\end{align*} by the total number of _____________.
5. Every square has an angle of rotation of _________.

Determine whether each statement is true or false.

1. Every parallelogram has rotational symmetry.
2. Every figure that has line symmetry also has rotational symmetry.

Determine whether the words below have rotation symmetry.

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

Find the angle of rotation and the number of times each figure can rotate.

Determine if the figures below have rotation symmetry. Identify the angle of rotation.

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### Vocabulary Language: English

TermDefinition
Center of Rotation In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.
Rotation A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.
Rotation Symmetry A figure has rotational symmetry if it can be rotated less than $360^\circ$ around its center point and look exactly the same as it did before the rotation.
Symmetry A figure has symmetry if it can be transformed and still look the same.