10.3: Rotations of Geometric Shapes
Objectives
The lesson objectives for Rotations of Geometric Shapes are:
- What is a rotation?
- Graphing an image that has undergone a rotation.
- Writing a rule for a rotation
- The properties of a rotation
What Is a Rotation?
Concept Content
As you learned in the previous lesson, there are four types of transformations. There, you learned about one of these transformations, reflections. The other three transformations are rotations, dilations and translations. In this lesson, you will learn about rotations. The transformations known as rotations involve turning or spinning a figure through a specific angle and direction and about a fixed point. This fixed point is called the turn center. A rotations is also called a turn and the turn center can be called the center of rotation. The figure below shows that the Preimage A is rotated \begin{align*}90^\circ\end{align*} counter-clockwise about the center point \begin{align*}A\end{align*} to form the rotated image. Point \begin{align*}A\end{align*} is the turn center.
In this concept you will learn to describe rotations and, in the process, learn what they are. You will also learn how to recognize rotations and to determine the center of rotation. In order to do this, you will need to know the following:
- A rotation about a point of a specific number of degrees that is positive (i.e. \begin{align*}90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CCW or counter-clockwise rotation.
- A rotation about a point of a specific number of degrees that is negative (i.e. \begin{align*}-90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CW or clockwise rotation.
- A rotation about the origin will require you to look at specific points in the preimage and compare them to their corresponding points in the rotated image. The table below shows how you would describe the different rotations about the origin.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*} (or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*} (or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*} (or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} |
In later concepts, you will also learn to graph rotations, write rules to describe them, and finally understand some of the properties of rotations.
Guidance
Which one of the following figures represents a rotation? Explain.
You know that a rotation is a transformation that turns or flips a figure about a fixed point. This fixed point is the turn center or the center of rotation. In the figures above, Figure 1 and Figure 3 involve turning the heart about a fixed point. Figure 1 turns the heart about the point \begin{align*}A\end{align*}. Point \begin{align*}A\end{align*} is the center of rotation. Figure 3 turns the heart about the point directly right of \begin{align*}A\end{align*}. This point is the center of rotation. So, Figure 1 and Figure 3 represent rotations.
Examples
Example A
Describe the rotation of the blue triangle in the diagram below.
Looking at the angle measures, \begin{align*}\angle ABA^\prime=90^\circ\end{align*}. Therefore the preimage, Image A, has been rotated \begin{align*}90^\circ\end{align*} about the point \begin{align*}B\end{align*}. Rotations are described as clockwise or counter-clockwise. Since you were to rotate an object \begin{align*}90^\circ\end{align*} about the Point \begin{align*}B\end{align*}, the triangle \begin{align*}ABC\end{align*} is said to be rotated \begin{align*}90^\circ\end{align*}CCW about the center of rotation Point \begin{align*}B\end{align*}. If you were to rotate an object \begin{align*}90^\circ\end{align*} to the left (or \begin{align*}-90^\circ\end{align*}), the rotation is said to be clockwise.
Example B
Describe the rotation of the triangles in the diagram below.
Looking at the angle measures, \begin{align*}\angle CAB^\prime + \angle B^\prime AC^\prime=180^\circ\end{align*}. Since you were to rotate an object \begin{align*}180^\circ\end{align*} about the Point \begin{align*}A\end{align*}, the triangle \begin{align*}ABC\end{align*} is said to be rotated \begin{align*}180^\circ\end{align*}CCW about the center of rotation Point \begin{align*}A\end{align*}.
Example C
Describe the rotation in the diagram below.
To describe the rotation in this diagram, look at the points indicated on the S shape.
Points \begin{align*}BC\end{align*} \begin{align*}B(-3, 4)\end{align*} \begin{align*}C(-5, 0)\end{align*}
Points \begin{align*}B^\prime C^\prime\end{align*} \begin{align*}B^\prime (4, 3)\end{align*} \begin{align*}C^\prime (0, 5)\end{align*}
From the table in the introduction, you know that these points represent a rotation of \begin{align*}90^\circ\end{align*}CW about the origin. Each coordinate point has followed the conditions of the \begin{align*}90^\circ\end{align*}CW meaning that the point \begin{align*}(x, y)\end{align*} becomes the point \begin{align*}(y, -x)\end{align*}.
Vocabulary
- Turn Center
- A turn center is the fixed point for which the figure is turned in a rotation. The turn center is also called the center of rotation.
- Rotations
- Rotations in transformations involve turning or spinning a figure through a specific angle and direction and about a fixed point.
Guided Practice
1. Describe the rotation of the pink triangle in the diagram below.
2. Describe the rotation of the blue polygon in the diagram below.
3. Describe the rotation of the green hexagon in the diagram below.
Answers
1. Examine the points of the preimage and the rotated image (the blue triangle).
Points on \begin{align*}BCD\end{align*} | \begin{align*}B(1, -1)\end{align*} | \begin{align*}C(2, 6)\end{align*} | \begin{align*}D(5, 1)\end{align*} |
Points on \begin{align*}B^\prime C^\prime D^\prime\end{align*} | \begin{align*}B^\prime (1, 1)\end{align*} | \begin{align*}C^\prime (-6, 2)\end{align*} | \begin{align*}D^\prime (-1, 5)\end{align*} |
From the table in the introduction, you know that these points represent a rotation of \begin{align*}90^\circ\end{align*}CCW about the origin. Each coordinate point has followed the conditions of the \begin{align*}90^\circ\end{align*}CCW meaning that the point \begin{align*}(x, y)\end{align*} becomes the point \begin{align*}(-y, x)\end{align*}.
2. For this image, look at the rotation. It is not rotated about the origin but rather about the point \begin{align*}A\end{align*}. We can measure the angle of rotation:
The blue polygon is being rotated about the point \begin{align*}A \ 145^\circ\end{align*} clockwise (or to the left). You would say that the blue polygon is rotated \begin{align*}145^\circ\end{align*}CW to form the orange polygon.
3. For this image, look at the rotation. It is not rotated about the origin but rather about the point \begin{align*}D\end{align*}. We can measure the angle of rotation:
The green polygon is being rotated about the point \begin{align*}D \ 90^\circ\end{align*} counter-clockwise (or to the right). You would say that the green hexagon is rotated \begin{align*}90^\circ\end{align*}CCW to form the orange hexagon.
Summary
In this concept you began your study of rotations, the third type of transformations. Remember rotations involve turning or spinning about a fixed point, including the origin. The rotation line is often called the mirror line. You learned the following types of rotations in this concept as well as how to describe them:
- A rotation about a point of a specific number of degrees that is positive (i.e. \begin{align*}90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CCW or counter-clockwise rotation.
- A rotation about a point of a specific number of degrees that is negative (i.e. \begin{align*}-90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CW or clockwise rotation.
- A rotation about the origin will require you to look at specific points in the preimage and compare them to their corresponding points in the rotated image. The table below shows how you would describe the different rotations about the origin.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*}(or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*}(or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*}(or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} |
In later concepts, you will also learn to graph rotations, write rules to describe them, and finally understand some of the properties of rotations.
Problem Set
If the following points were rotated about the origin with a \begin{align*}180^\circ\end{align*}CCW rotation, what would be the coordinates of the rotated points? Show these rotations on a graph.
- (3, 1)
- (4, –2)
- (–5, 3)
- (–6, 4)
- (–3, –3)
If the following points were rotated about the origin with a \begin{align*}90^\circ\end{align*}CW rotation, what would be the coordinates of the rotated points? Show these rotations on a graph.
- (–4, 3)
- (5, –4)
- (–5, –4)
- (3, 3)
- (–8, –9)
Describe the following rotations:
Graphing an Image that has Undergone a Rotation
Concept Content
In this second concept of lesson What is a Rotation?, you will learn how to graph an image that has undergone a rotation. Remember that a rotation is when a preimage is turned or spun through a specific angle and direction and about a fixed point.
In the last lesson you learned to recognize what happens to the points and images when rotated in different directions and angles. In this lesson you will create the images and preimages by first graphing them on a Cartesian plane. When graphing the rotated image, it is often helpful to remember what happens to the points in the preimage depends on the type of rotation. The table below summarizes the types of rotations and what happens to the points in your preimage based on each type.
- A rotation about a point of a specific degrees that is positive (i.e. \begin{align*}90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CCW or counter-clockwise rotation.
- A rotation about a point of a specific degrees that is negative (i.e. \begin{align*}-90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CW or clockwise rotation.
- A rotation about the origin will require you to look at specific points in the preimage and compare them to their corresponding points in the rotated image. The table below shows how you would describe the different rotations about the origin.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*}(or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*}(or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*}(or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} |
Guidance
Quadrilateral \begin{align*}WXYZ\end{align*} has coordinates \begin{align*}W(-5, 5), X(-2, 0), Y(2, 3)\end{align*} and \begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane. Rotate the image \begin{align*}110^\circ\end{align*} clockwise about the point \begin{align*}X\end{align*}. Show the resulting image.
Examples
Example A
Line \begin{align*}\overline{AB}\end{align*} drawn from (–4, 2) to (3, 2) has been rotated about the origin at an angle of \begin{align*}90^\circ\end{align*}CW. Draw the preimage and image and properly label each.
Example B
The diamond \begin{align*}ABCD\end{align*} is rotated \begin{align*}145^\circ\end{align*}CCW about the origin to form the image \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}. Find the coordinates of the rotated image. On the diagram, draw and label the reflected image.
Notice the direction of the angle is counter-clockwise.
Example C
The following figure is rotated about the origin \begin{align*}200^\circ\end{align*}CW to make a rotated image. Find the coordinates of the rotated image. On the diagram, draw and label the image.
Notice the direction of the angle is counter-clockwise, therefore the angle measure is \begin{align*}160^\circ\end{align*}.
Vocabulary
- Turn Center
- A turn center is the fixed point for which the figure is turned in a rotation. The turn center is also called the center of rotation.
- Rotations
- Rotations in transformations involve turning or spinning a figure through a specific angle and direction and about a fixed point.
Guided Practice
1. Line \begin{align*}\overline{ST}\end{align*} drawn from (–3, 4) to (–3, 8) has been rotated \begin{align*}60^\circ\end{align*}CW about the point \begin{align*}S\end{align*}. Draw the preimage and image and properly label each.
2. The polygon below has been rotated \begin{align*}155^\circ\end{align*}CCW about the origin. Draw the rotated image and properly label each.
3. The purple pentagon is rotated about the point \begin{align*}A \ 225^\circ\end{align*}. Find the coordinates of the purple pentagon. On the diagram, draw and label the rotated pentagon.
Answers
1.
Notice the direction of the angle is clockwise, therefore the angle measure is \begin{align*}60^\circ\end{align*}CW or \begin{align*}-60^\circ\end{align*}.
2.
Notice the direction of the angle is counter-clockwise, therefore the angle measure is \begin{align*}155^\circ\end{align*}CCW or \begin{align*}155^\circ\end{align*}.
3.
The measure of \begin{align*}\angle BAB^\prime = m \angle BAE^\prime + m \angle E^\prime AB^\prime\end{align*}. Therefore \begin{align*}\angle BAB^\prime = 111.80^\circ + 113.20^\circ\end{align*} or \begin{align*}225^\circ\end{align*}. Notice the direction of the angle is counter-clockwise, therefore the angle measure is \begin{align*}225^\circ\end{align*}CCW or \begin{align*}225^\circ\end{align*}.
Summary
In this third concept of Chapter Is it a Slide, a Flip, or a Turn? you were introduced to describing rotations, keeping in mind the way the angle rotates about a fixed point or the origin. In this lesson you worked from the rotation description and graphed the rotated image. You used the following concepts to graph the rotated images:
- A rotation about a point of a specific number of degrees that is positive (i.e. \begin{align*}90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CCW or counter-clockwise rotation.
- A rotation about a point of a specific number of degrees that is negative (i.e. \begin{align*}-90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CW or clockwise rotation.
- A rotation about the origin will require you to look at specific points in the preimage and compare them to their corresponding points in the rotated image. The table below shows how you would describe the different rotations about the origin.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*}(or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*}(or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*}(or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} |
Problem Set
- Rotate figure \begin{align*}MATH\end{align*} on the grid about the origin \begin{align*}180^\circ\end{align*}.
- Rotate figure \begin{align*}DEFGH\end{align*} about the origin \begin{align*}-270^\circ\end{align*}.
- Rotate figure \begin{align*}LMO\end{align*} about the point \begin{align*}M\end{align*} by \begin{align*}-45^\circ\end{align*}.
- Rotate figure \begin{align*}WXY\end{align*} about the point \begin{align*}W\end{align*} by \begin{align*}-45^\circ\end{align*}.
- Rotate the purple figure about the origin by \begin{align*}-90^\circ\end{align*}.
For each of the following diagrams rotate the images about the origin by \begin{align*}90^\circ\end{align*}CCW. Find the coordinates of the preimages. On each diagram, draw and label the reflected image.
For each of the following diagrams rotate the images \begin{align*}60^\circ\end{align*}CW about one of the points. Find the coordinates of the preimages. On each diagram, draw and label the rotated image.
Writing a Rule for a Rotation
Concept Content
In this third concept with respect to rotations, you will learn how to describe a rotation using a rule. In the first lesson you used words to describe a rotation. By examining the coordinates of the rotated image, you could describe if the preimage was rotated about the origin with a specific direction and a specific angle. You will focus on only rotations about the origin in this concept as the notation rules are specific to those found in the table below.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) | Notation (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*}(or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} | \begin{align*}(x, y) \rightarrow (-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*}(or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} | \begin{align*}(x, y) \rightarrow (-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*}(or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} | \begin{align*}(x, y) \rightarrow (y, -x)\end{align*} |
If, for instance you had the image below:
You would notice that the preimage is rotated in the origin \begin{align*}90^\circ\end{align*}CCW. If you were to describe the rotated image using notation, you would write the following:
\begin{align*}(x,y) \rightarrow (-y,x) \ \text{or} \ R_{90^\circ}(x,y)=(-y,x)\end{align*}
In this concept you will write rules for rotation as well as draw graphs from these rules. For this concept you will be using the second of the above notations, namely \begin{align*}R_{90^\circ}(x,y)=(-y,x)\end{align*}.
Guidance
The figure below shows a pattern of two fish. Write the mapping rule for the rotation of Image A to Image B.
Notice that the angle measure is \begin{align*}90^\circ\end{align*} and the direction is clockwise. Therefore the Image A has been rotated \begin{align*}-90^\circ\end{align*} to form Image B. To write a rule for this rotation you would write: \begin{align*}R_{270^\circ}(x,y)=(y,-x)\end{align*}.
Examples
Example A
Find an image of the point (3, 2) that has undergone a rotation in:
a) about the origin at \begin{align*}90^\circ\end{align*},
b) about the origin at \begin{align*}180^\circ\end{align*}, and
c) about the origin at \begin{align*}270^\circ\end{align*}.
Write the notation to describe the rotation.
a) Rotation about the origin at \begin{align*}90^\circ : R_{90^\circ}(x,y)=(-y,x)\end{align*}
b) Rotation about the origin at \begin{align*}180^\circ : R_{180^\circ}(x,y)=(-x,-y)\end{align*}
c) Rotation about the origin at \begin{align*}270^\circ : R_{270^\circ}(x,y)=(y,-x)\end{align*}
Example B
Rotate Image A in the diagram below:
d) about the origin at \begin{align*}90^\circ\end{align*}, and label it \begin{align*}B\end{align*}.
e) about the origin at \begin{align*}180^\circ\end{align*}, and label it \begin{align*}O\end{align*}.
f) about the origin at \begin{align*}270^\circ\end{align*}, and label it \begin{align*}Z\end{align*}.
Write notation for each to indicate the type of rotation.
a) Rotation about the origin at \begin{align*}90^\circ\end{align*}: \begin{align*}R_{90^\circ}A \rightarrow B=R_{90^\circ}(x,y) \rightarrow (-y,x)\end{align*}
b) Rotation about the origin at \begin{align*}180^\circ\end{align*}: \begin{align*}R_{180^\circ}A \rightarrow O=R_{180^\circ}(x,y) \rightarrow (-x,-y)\end{align*}
c) Rotation about the origin at \begin{align*}270^\circ\end{align*}: \begin{align*}R_{270^\circ}A \rightarrow Z=R_{270^\circ}(x,y) \rightarrow (y,-x)\end{align*}
Example C
Write the notation that represents the rotation of the preimage A to the rotated image J in the diagram below.
First, pick a point in the diagram to use to see how it is rotated.
\begin{align*}E: (-1, 2) \quad E^\prime: (1, -2)\end{align*}
Notice how both the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are multiplied by –1. This indicates that the preimage A is rotated about the origin by \begin{align*}180^\circ\end{align*}CCW to form the rotated image J. Therefore the notation is \begin{align*}R_{180^\circ}A \rightarrow J=R_{180^\circ}(x,y) \rightarrow (-x,-y)\end{align*}.
Vocabulary
- Notation Rule
- A notation rule has the following form \begin{align*}R_{180^\circ}A \rightarrow O=R_{180^\circ}(x,y) \rightarrow (-x,-y)\end{align*} and tells you that the image A has been rotated about the origin and both the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are multiplied by –1.
- Rotations
- Rotations in transformations involve turning or spinning a figure through a specific angle and direction and about a fixed point.
Guided Practice
1. Thomas describes a translation as point \begin{align*}J\end{align*} moving from \begin{align*}J(-2, 6)\end{align*} to \begin{align*}J^\prime(6, 2)\end{align*}. Write the notation to describe this rotation for Thomas.
2. Write the notation that represents the rotation of the yellow diamond to the rotated green diamond in the diagram below.
3. Karen was playing around with a drawing program on her computer. She created the following diagrams and then wanted to determine the transformations. Write the notation rule that represents the transformation of the purple and blue diagram to the orange and blue diagram.
Answers
1. \begin{align*}J: (-2, 6) \quad J^\prime: (6, 2)\end{align*}
Since the \begin{align*}x\end{align*}-coordinate is multiplied by –1, the \begin{align*}y\end{align*}-coordinate remains the same, and finally the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates change places, this is a rotation about the origin by \begin{align*}270^\circ\end{align*} or \begin{align*}-90^\circ\end{align*}. The notation is: \begin{align*}R_{270^\circ}J \rightarrow J^\prime=R_{270^\circ}(x,y) \rightarrow (y,-x)\end{align*}
2. In order to write the notation to describe the rotation, choose one point on the preimage (the yellow diamond) and then the rotated point on the green diamond to see how the point has moved. Notice that point \begin{align*}E\end{align*} is shown in the diagram:
\begin{align*}E(-1,3) \rightarrow E^\prime(-3,-1)\end{align*}
Since both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are reversed places and the \begin{align*}y\end{align*}-coordinate has been multiplied by –1, the rotation is about the origin \begin{align*}90^\circ\end{align*}. The notation for this rotation would be: \begin{align*}R_{90^\circ}(x,y) \rightarrow (-y,x)\end{align*}.
3. In order to write the notation to describe the transformation, choose one point on the preimage (purple and blue diagram) and then the transformed point on the orange and blue diagram to see how the point has moved. Notice that point \begin{align*}C\end{align*} is shown in the diagram:
\begin{align*}C(7,0) \rightarrow C^\prime(0,-7)\end{align*}
Since the \begin{align*}x\end{align*}-coordinates only are multiplied by –1, and then \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates change places, the transformation is a rotation is about the origin by \begin{align*}270^\circ\end{align*}. The notation for this rotation would be: \begin{align*}R_{270^\circ}(x,y) \rightarrow (y,-x)\end{align*}.
Summary
In this concept of lesson Rotations of Geometric Shapes you have learned to describe a rotation in notation form. Notations for rotations can allow you to quickly determine at what angle and direction the rotation occurs as well as help you determine the coordinates of the rotated points. In other words, you can draw a rotated image from a preimage knowing the notation. In order to write a notation you use the following general forms:
Center of Rotation | Angle of Rotation | Notation rule |
---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*} (or \begin{align*}-270^\circ\end{align*}) | \begin{align*}R_{90^\circ}(x,y)=(-y,x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*} (or \begin{align*}-180^\circ\end{align*}) | \begin{align*}R_{180^\circ}(x,y) \rightarrow (-x,-y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*} (or \begin{align*}-90^\circ\end{align*}) | \begin{align*}R_{270^\circ}(x,y) \rightarrow (y,-x)\end{align*} |
Using these mapping rules not only allows you to quickly determine the types of rotations in images but also allows you to draw a rotated image from the notation. Remember as well that a positive angle of rotation turns the figure counterclockwise, and a negative angle of rotation turns the figure in a clockwise direction.
Problem Set
Complete the following table:
Starting Point | \begin{align*}90^\circ\end{align*} Rotation | \begin{align*}180^\circ\end{align*} Rotation | \begin{align*}270^\circ\end{align*} Rotation | \begin{align*}360^\circ\end{align*} Rotation |
---|---|---|---|---|
1. (1, 4) | ||||
2. (4, 2) | ||||
3. (2, 0) | ||||
4. (–1, 2) | ||||
5. (–2, –3) |
Write the notation that represents the rotation of the preimage A to the reflected images in the diagrams below.
The Properties of a Rotation
Concept Content
As you learned earlier, rotations involve turns or spins of a preimage to form a rotated image. In this last concept you will look at the properties of rotations. Rotations result when an image and its rotated image have the same shape but the rotated image may be turned in different directions. You have seen in the examples of rotations from the previous three concepts where a rotation involves turning or spinning an image across the origin or from a point on the preimage. Therefore rotated images have the same length and angle measurements as their preimages. As well, points of the rotated image are collinear if they are collinear in the preimage. In other words, if \begin{align*}B\end{align*} is between \begin{align*}A\end{align*} and \begin{align*}C\end{align*}, then \begin{align*}B^\prime\end{align*} will be between \begin{align*}A^\prime\end{align*} and \begin{align*}C^\prime\end{align*}.
In summary, rotations are considered to have the following properties:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- same orientation (lettering remains the same)
- points that are on each line remain on the line for the reflected image (collinear)
- parallelism (parallel lines remain parallel)
Guidance
The triangle \begin{align*}ABC\end{align*} is drawn such that the vertices are at \begin{align*}A(1, 1), B(5, 5)\end{align*} and \begin{align*}C(-2, 4)\end{align*}. Triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*} is rotated \begin{align*}45^\circ\end{align*}CW.
a) What are the coordinates of Triangle \begin{align*}A^\prime B^\prime C^\prime \end{align*}?
b) Measure each angle either with a protractor or using geometry software. What angles are congruent?
c) Measure each side length. Are the distances the same?
d) Can you conclude that the triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*} is a rotation of triangle \begin{align*}ABC\end{align*}?
a) The coordinates of \begin{align*}A^\prime B^\prime C^\prime\end{align*} upon rotation \begin{align*}45^\circ\end{align*}CW would be:
Preimage \begin{align*}ABC\end{align*} \begin{align*}A(1, 1)\end{align*} \begin{align*}B(5, 5)\end{align*} \begin{align*}C(-2, 4)\end{align*}
Reflected Image \begin{align*}A^\prime B^\prime C^\prime\end{align*} \begin{align*}A^\prime (-0.7, 5)\end{align*} \begin{align*}B^\prime (5, 5)\end{align*} \begin{align*}C^\prime (-0.7, -9.2)\end{align*}
b) The measures of each angle are shown in the diagram.
Congruent angles have the same measure. Since \begin{align*}\angle A= \angle A^\prime, \angle B=\angle B^\prime\end{align*}, and \begin{align*}\angle C= \angle C^\prime\end{align*}, the angles are congruent.
c)
The side lengths are also the same \begin{align*}(m \overline{AC}=m \overline{A^\prime C^\prime},m \overline{BC}=m \overline{B^\prime C^\prime}\end{align*}, and \begin{align*}m \overline{AB}=m \overline{A^\prime C^\prime})\end{align*}.
d) Since the angle measures are the same and the lengths of each side are the same, the triangles are congruent. Triangle \begin{align*}ABC\end{align*} has been reflected to produce triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}.
Examples
Example A
Graph the following two squares to determine if Quadrilateral A is rotated to form Quadrilateral B.
Quadrilateral A Vertices: \begin{align*}W(2, -2), X(5, 0), Y(5, 3),\end{align*} and \begin{align*}Z(4, 3)\end{align*}.
Quadrilateral B Vertices: \begin{align*}W^\prime (2, 2), X^\prime (0, 5), Y^\prime (-3, 5),\end{align*} and \begin{align*}Z^\prime (-3, 4)\end{align*}.
The side lengths are the same (\begin{align*}m \overline{WX}=m \overline{W^\prime X^\prime},m \overline{WZ}=m \overline{W^\prime Z^\prime},m \overline{YZ}=m \overline{Y^\prime Z^\prime}\end{align*}, and \begin{align*}m \overline{XY}=m \overline{X^\prime Y^\prime}\end{align*}) and the angle lengths are the same \begin{align*}(\angle X= \angle X^\prime, \angle Y=\angle Y^\prime, \angle W= \angle W^\prime\end{align*}, and \begin{align*}\angle Z= \angle Z^\prime)\end{align*}.
Since the side and angle measures are the same Quadrilateral A is congruent to Quadrilateral B. Each y-coordinate on \begin{align*}ABCD\end{align*} is multiplied by –1 and then coordinate points are reversed to produce Quadrilateral B. With all of this, you can conclude that Quadrilateral B is a rotated image of the preimage Quadrilateral A. The rotation is about the origin \begin{align*}90^\circ\end{align*}CCW.
Example B
Describe how you would know if image \begin{align*}T\end{align*} is rotated onto image \begin{align*}T^\prime\end{align*}.
As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image \begin{align*}T\end{align*} is congruent to Image \begin{align*}T^\prime\end{align*}. Each \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate is multiplied by –1. With all of this, you can conclude that the Image \begin{align*}T\end{align*} is rotated about the origin \begin{align*}180^\circ\end{align*} to form Image \begin{align*}T^\prime\end{align*}.
Example C
Describe how you would know if image \begin{align*}R\end{align*} is rotated to form image \begin{align*}R^\prime\end{align*}.
As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image \begin{align*}R\end{align*} is congruent to Image \begin{align*}R^\prime\end{align*}. Looking at point \begin{align*}E\end{align*} and \begin{align*}E^\prime\end{align*}, the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates have been reversed and then the \begin{align*}x\end{align*}-coordinate has been multiplied by –1. With all of this, you can conclude that Image \begin{align*}R\end{align*} is rotated across the origin at \begin{align*}90^\circ\end{align*}CW to form the image \begin{align*}R^\prime\end{align*}.
Vocabulary
- Congruent Angles
- Congruent angles have the same measure.
- Rotations
- Rotations in transformations involve turning or spinning a figure through a specific angle and direction and about a fixed point.
Guided Practice
1. Describe how you would know if image A is rotated onto image B.
2. Describe how you would know if image S is rotated onto image T.
3. Is the red octagon a rotated image of the purple octagon? How do you know?
Answers
1.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image A is congruent to Image B. Each \begin{align*}x\end{align*}-coordinate point is multiplied by –1 while each \begin{align*}y\end{align*}-coordinate value remains the same. With all of this, you can conclude that Image B is a rotated image of the preimage Image A. Preimage A is in fact rotated across the origin at \begin{align*}90^\circ\end{align*}CW to form Image B.
2.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image S is congruent to Image T. Each \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate has been multiplied by –1. With all of this, you can conclude that Image T is a rotated image of the Preimage S. As well, Preimage S has been rotated across the origin at \begin{align*}180^\circ\end{align*} in order to produce Image T.
3.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image J is congruent to Image \begin{align*}J^\prime\end{align*}. Each \begin{align*}y\end{align*}-coordinate has been multiplied by –1. With all of this, you can conclude that Image \begin{align*}J^\prime\end{align*} is a rotated image of the Preimage J. As well, Preimage J has been rotated across the origin at \begin{align*}270^\circ\end{align*}CCW or \begin{align*}90^\circ\end{align*}CW in order to produce Image \begin{align*}J^\prime\end{align*}.
Summary
The previous three lessons you worked with rotations, a transformation of a figure that involves turning a point or a figure along the Cartesian plane. In this lesson you looked more closely at the properties of rotations. These properties include:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- same orientation (lettering remains the same)
- points that are on each line remain on the line for the reflected image (collinear)
- parallelism (parallel lines remain parallel)
Through the use of geometry software, it is easy to measure both the angles and the side lengths to determine if these elements are congruent, thus satisfying the first two properties. If points are each moving in the same direction and are at the same distance measure, you can be sure that the preimage is rotated to form the new image.
Problem Set
Describe how you would know if image A is rotated onto image B in each of the following diagrams.
- Triangle \begin{align*}ABC\end{align*} with vertices \begin{align*}A(3, 1), B(0, -1)\end{align*} and \begin{align*}C(1, 5)\end{align*} has been rotated about the point \begin{align*}P(-1, 1) \ 180^\circ\end{align*} to produce triangle \begin{align*}A^\prime B^\prime C^\prime\end{align*}. Draw the diagram representing this rotation and prove the image \begin{align*}A^\prime B^\prime C^\prime\end{align*} is actually a rotation.
- Triangle \begin{align*}DEF\end{align*} with vertices \begin{align*}D(0, 1), E(5, 1)\end{align*} and \begin{align*}F(2, 5)\end{align*} has been rotated about the point \begin{align*}P(-2, 5) \ 90^\circ\end{align*}CCW to produce triangle \begin{align*}D^\prime E^\prime F^\prime\end{align*}. Draw the diagram representing this rotation and prove the image \begin{align*}D^\prime E^\prime F^\prime\end{align*} is actually a rotation.
- Draw a letter from the alphabet that produces the same letter after a \begin{align*}180^\circ\end{align*} rotation. Prove the rotated image is actually a rotation.
- Draw another letter from the alphabet that produces the same letter after a \begin{align*}180^\circ\end{align*} rotation. Prove the rotated image is actually a rotation.
- Prove that the preimage triangle 1 is rotated to produce triangle 2. Determine the rotation of triangle 1 to triangle 2.
Summary
In this third lesson of Chapter Is it a Slide, a Flip, or a Turn? you have been introduced to rotations. Remember that a rotation is just a turn or spin of the preimage to create a rotated image. Rotations are an important concept as you notice them probably daily in your everyday life. Amusement park rides such as the Ferris wheel and the tilt-a-whirl involve rotations (of you), blades on a ceiling fan involve rotations, and even some flags use rotations to create the pattern.
You first learned to describe the rotation using words. So, you learned to look at points in a rotated image and the preimage to determine if the images involve:
- a rotation about a point clockwise,
- a rotation about a point counter-clockwise, or
- a rotation about the origin of a specific angle and direction
In this lesson you described the type of rotation using words. Following this you learned to draw a rotated image based on a description. You further identified the types of rotations using the following data.
- A rotation about a point of a specific number of degrees that is positive (i.e. \begin{align*}90\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CCW or counter-clockwise rotation.
- A rotation about a point of a specific number of degrees that is negative (i.e. \begin{align*}-90^\circ\end{align*}) is described as a \begin{align*}90^\circ\end{align*}CW or clockwise rotation.
- A rotation about the origin will require you to look at specific points in the preimage and compare them to their corresponding points in the rotated image. The table below shows how you would describe the different rotations about the origin.
Center of Rotation (turn point) | Angle of Rotation | Preimage (Point \begin{align*}P\end{align*}) | Rotated Image (Point \begin{align*}P^\prime\end{align*}) |
---|---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*} (or \begin{align*}-270^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*} (or \begin{align*}-180^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*} (or \begin{align*}-90^\circ\end{align*}) | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, -x)\end{align*} |
Therefore if you were asked to rotate an image about the origin at \begin{align*}90^\circ\end{align*}CCW, you quickly knew to multiply the \begin{align*}y\end{align*}-coordinate by –1 and then to interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates of the preimage to form the rotated image.
In the third concept, you learned to write the notation for the rotated image from a preimage. You learned to graph a rotated image and then write the notation for the image. Your new table to use when working with rotations became:
Center of Rotation | Angle of Rotation | Notation rule |
---|---|---|
(0, 0) | \begin{align*}90^\circ\end{align*} (or \begin{align*}-270^\circ\end{align*}) | \begin{align*}R_{90^\circ}(x,y)=(-y,x)\end{align*} |
(0, 0) | \begin{align*}180^\circ\end{align*} (or \begin{align*}-180^\circ\end{align*}) | \begin{align*}R_{180^\circ}(x,y) \rightarrow (-x,-y)\end{align*} |
(0, 0) | \begin{align*}270^\circ\end{align*} (or \begin{align*}-90^\circ\end{align*}) | \begin{align*}R_{270^\circ}(x,y) \rightarrow (y,-x)\end{align*} |
Finally in the last concept of this lesson, you learned that rotations have the following properties:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- same orientation (lettering remains the same)
- points that are on each line remain on the line for the reflected image (collinear)
- parallelism (parallel lines remain parallel)
You used these tendencies to prove that one image is a rotation of another image by finding the lengths of the preimage and the rotated image as well as the angle measures in both images. You then found where the points moved from the preimage to the rotated image. All of these properties combined proved that one image was a rotation of the other.
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