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Rotations

Transformations by which a figure is turned around a fixed point to create an image.

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Recognizing Rotation Transformations
License: CC BY-NC 3.0

This smiley face has a number of distinct features, but does it have rotational symmetry?

In this concept, you will learn to recognize rotation transformations and rotational symmetry.

Rotation

A transformation is the movement of a geometric figure on the coordinate plane. A rotation is a type of transformation which is a turn. A figure can be turned clockwise or counterclockwise on the coordinate plane. In both transformations the size and shape of the figure stays exactly the same.

A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction. You can turn a figure 90°, a quarter turn, either clockwise or counterclockwise. When you spin the figure exactly halfway, you have rotated it 180°. Turning it all the way around rotates the figure 360°.

Look at the triangles below. The pink triangle is a 90° clockwise rotation or is a quarter turn.

License: CC BY-NC 3.0

You could also turn the blue triangle 180°, which would turn the pink triangle completely upside down.

License: CC BY-NC 3.0

Next, let’s look at an example of rotating figures on the coordinate plane.

Rotate this figure 90° clockwise on the coordinate plane.

License: CC BY-NC 3.0

First, write down the coordinates for each of the points of this pentagon.

\begin{align*}A (-3, 5)\end{align*}A(3,5)

\begin{align*}B (-4, 4)\end{align*}B(4,4)

\begin{align*}C (-3, 3)\end{align*}C(3,3)

\begin{align*}D(-1,2)\end{align*}D(1,2)

\begin{align*}E (-1, 4)\end{align*}E(1,4)

Next, the easiest way to think about rotating any figure is to think about it moving around a fixed point. In the case of graphing figures on the coordinate plane, you will be rotating the figures around the center point or origin.

If you rotate a figure clockwise 90°, then you are going to be shifting the whole figure along the \begin{align*}x\end{align*}x-axis.

To figure out the coordinates of the new rotated figure, you switch the coordinates and then, you need to multiply the second coordinate by -1. This will make perfect sense given that the entire figure is going to shift.

\begin{align*}\begin{array}{rcl} && \text{Original Points} \qquad \text{Coordinates Switched} \qquad y \text{-coordinate multiplied by} -1\\ && \qquad A (-3, 5) \qquad \qquad \quad \quad A^t (5, -3) \qquad \qquad \qquad \qquad \quad A^t(5, 3) \\ && \qquad B (-4, 4) \qquad \qquad \quad \quad B^t (4, -4) \qquad \qquad \qquad \qquad \quad B^t(4, 4) \\ && \qquad C (-3, 3) \qquad \qquad \quad \quad C^t (3, -3) \qquad \qquad \qquad \qquad \quad C^t(3, 3) \\ && \qquad D (-1, 2) \qquad \qquad \quad \quad D^t (2, -1) \qquad \qquad \qquad \qquad \quad D^t(2, 1) \\ && \qquad E (-1, 4) \qquad \qquad \quad \quad E^t (4, -1) \qquad \qquad \qquad \qquad \quad E^t(4, 1) \end{array}\end{align*}Original PointsCoordinates Switchedy-coordinate multiplied by1A(3,5)At(5,3)At(5,3)B(4,4)Bt(4,4)Bt(4,4)C(3,3)Ct(3,3)Ct(3,3)D(1,2)Dt(2,1)Dt(2,1)E(1,4)Et(4,1)Et(4,1)

Then, graph this rotated figure. Notice that you use \begin{align*}A^t\end{align*}At to represent the rotated figure.

License: CC BY-NC 3.0

Let’s think about what would happen to the figure if you were to rotate it counterclockwise. To do this, the figure would move across the \begin{align*}y\end{align*}y-axis in fact, the \begin{align*}x\end{align*}x-coordinates would change completely. In actuality, you would switch the original coordinates around. The \begin{align*}x\end{align*}x-coordinate would become the \begin{align*}y\end{align*}y-coordinate and the \begin{align*}y\end{align*}y-coordinate would become the \begin{align*}x\end{align*}x-coordinate. Then, you need to multiply the new \begin{align*}x\end{align*}-coordinate by -1.

Let’s take a look.

\begin{align*}\begin{array}{rcl} && \text{Original Points} \qquad \text{Coordinates Switched} \qquad x \text{-coordinate multiplied by} -1\\ && \qquad A (-3, 5) \qquad \qquad \quad \quad A^t (5, -3) \qquad \qquad \qquad \qquad A^t(-5, -3) \\ && \qquad B (-4, 4) \qquad \qquad \quad \quad B^t (4, -4) \qquad \qquad \qquad \qquad B^t(-4, -4) \\ && \qquad C (-3, 3) \qquad \qquad \quad \quad C^t (3, -3) \qquad \qquad \qquad \qquad C^t(-3, -3) \\ && \qquad D (-1, 2) \qquad \qquad \quad \quad D^t (2, -1) \qquad \qquad \qquad \qquad D^t(-2, -1) \\ && \qquad E (-1, 4) \qquad \qquad \quad \quad E^t (4, -1) \qquad \qquad \qquad \qquad E^t(-4, -1) \end{array}\end{align*}

Let’s graph this new counterclockwise rotation.

License: CC BY-NC 3.0

You can also graph figures that have been rotated 180°. To do this, you multiply both coordinates of the original figure by -1.

Let’s see what this looks like.

\begin{align*}\begin{array}{rcl} && \text{Original Points} \qquad \qquad \ \text{Coordinates multiplied by} -1\\ && \qquad A (-3, 5) \qquad \qquad \quad \quad A^t (3, -5) \\ && \qquad B (-4, 4) \qquad \qquad \quad \quad B^t (4, -4) \\ && \qquad C (-3, 3) \qquad \qquad \quad \quad C^t (3, -3) \\ && \qquad D (-1, 2) \qquad \qquad \quad \quad D^t (1, -2) \\ && \qquad E (-1, 4) \qquad \qquad \quad \quad E^t (1, -4) \end{array}\end{align*}

Let’s graph this 180° rotated image.

License: CC BY-NC 3.0

A figure has rotational symmetry if, when you rotate it, the figure appears to stay the same. The outlines do not change even as the figure turns.

Let’s look at an example.

License: CC BY-NC 3.0

The star will look the same even if you rotate it. You could turn it 72° or 144° clockwise or counterclockwise, it won’t matter. The star will still appear the same.

Examples

Example 1


Earlier, you were given a problem about the smiley face. How can you determine if it has rotational symmetry?

First, look at the outline of the image. The outline is a circle. A circle is the one shape that can be rotated less than 360° and still appears exactly the same. The circle is infinitely symmetrical because it keeps the shape no matter how many degrees it is moved or rotated.

Next, look at the design inside the image. If you rotate the circle, then the design inside will change.

The answer is that the image does not have rotational symmetry.

Example 2

Does a regular hexagon have rotational symmetry?

Look at the image of the regular hexagon below.

License: CC BY-NC 3.0

It has rotational symmetry. You can see that because you can rotate it 90° and 180° and it will still look exactly the same. You could rotate it less than 90° too and it still has rotational symmetry. You can also look at the angles to determine rotational symmetry. Each time we turn the figure, it has two parallel sides on the top and bottom and four other sides at the same angles. It has rotational symmetry.

Example 3

Does a square have rotational symmetry?

Look at the image of the square below.

License: CC BY-NC 3.0

Yes, the square has rotational symmetry because you can turn it and it will appear exactly the same.

Example 4

Does the letter U have rotational symmetry?

Look at the image of the letter U below.

License: CC BY-NC 3.0

No, the letter U does not have rotational symmetry because it will not appear the same if it is turned 90° clockwise or counterclockwise, or turned 180°.

Example 5

Does a regular octagon have a rotational symmetry?

Look at the image of the regular octagon below.

License: CC BY-NC 3.0

Yes, the regular octagon has rotational symmetry because you can turn it and it will appear exactly the same.

Review

Answer the following questions about rotations, translations and tessellations.

1. What is a translation?

2. What is a rotation?

3. What is a tessellation?

4. True or false. A figure can be translated up or down only.

5. True or false. A figure can be translated 180°.

6. True or false. A figure can be rotated 90° clockwise or counterclockwise.

7. True or false. A figure can’t be translated 180°.

8. When rotating a figure 90° counterclockwise, we switch the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} coordinates and multiply which one by -1?

9. When rotating a figure 90° clockwise, we multiply which coordinate by -1?

10. True or false. When rotating a figure 180°, we multiply both coordinates by -1.

Write the new coordinates for each rotation given the directions.

A Triangle with the coordinates (-4, 4) (-4, 2) and (-1, 1).

11. Rotate the figure 90° clockwise.

12. Rotate the figure 90° counterclockwise.

13. Rotate the figure 180°.

A Triangle with the coordinates (1, 3) (5, 1) (5, 3).

14. Rotate the figure clockwise 90°.

15. Rotate the figure counterclockwise 90°.

16. Rotate the figure 180°.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.15. 

Resources

Vocabulary

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.

Coordinate Notation

A coordinate point is the description of a location on the coordinate plane. Coordinate points are commonly written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.

Image

The image is the final appearance of a figure after a transformation operation.

Origin

The origin is the point of intersection of the x and y axes on the Cartesian plane. The coordinates of the origin are (0, 0).

Preimage

The pre-image is the original appearance of a figure in a transformation operation.

Reflection

A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.

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.

Rotational 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.

Transformation

A transformation moves a figure in some way on the coordinate plane.

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0
  7. [7]^ License: CC BY-NC 3.0
  8. [8]^ License: CC BY-NC 3.0
  9. [9]^ License: CC BY-NC 3.0
  10. [10]^ License: CC BY-NC 3.0
  11. [11]^ License: CC BY-NC 3.0
  12. [12]^ License: CC BY-NC 3.0

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