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10.3: Rotations of Geometric Shapes

Difficulty Level: At Grade Created by: CK-12

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 90^\circ counter-clockwise about the center point A to form the rotated image. Point A 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:

  1. A rotation about a point of a specific number of degrees that is positive (i.e. 90^\circ) is described as a 90^\circCCW or counter-clockwise rotation.
  2. A rotation about a point of a specific number of degrees that is negative (i.e. -90^\circ) is described as a 90^\circCW or clockwise rotation.
  3. 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 P) Rotated Image (Point P^\prime)
(0, 0) 90^\circ (or -270^\circ) (x, y) (-y, x)
(0, 0) 180^\circ (or -180^\circ) (x, y) (-x, -y)
(0, 0) 270^\circ (or -90^\circ) (x, y) (y, -x)

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 A. Point A is the center of rotation. Figure 3 turns the heart about the point directly right of A. 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, \angle ABA^\prime=90^\circ. Therefore the preimage, Image A, has been rotated 90^\circ about the point B. Rotations are described as clockwise or counter-clockwise. Since you were to rotate an object 90^\circ about the Point B, the triangle ABC is said to be rotated 90^\circCCW about the center of rotation Point B. If you were to rotate an object 90^\circ to the left (or -90^\circ), the rotation is said to be clockwise.

Example B

Describe the rotation of the triangles in the diagram below.

Looking at the angle measures, \angle CAB^\prime + \angle B^\prime AC^\prime=180^\circ. Since you were to rotate an object 180^\circ about the Point A, the triangle ABC is said to be rotated 180^\circCCW about the center of rotation Point A.

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 BC B(-3, 4) C(-5, 0)

Points B^\prime C^\prime B^\prime (4, 3) C^\prime (0, 5)

From the table in the introduction, you know that these points represent a rotation of 90^\circCW about the origin. Each coordinate point has followed the conditions of the 90^\circCW meaning that the point (x, y) becomes the point (y, -x).

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 BCD B(1, -1) C(2, 6) D(5, 1)
Points on B^\prime C^\prime D^\prime B^\prime (1, 1) C^\prime (-6, 2) D^\prime (-1, 5)

From the table in the introduction, you know that these points represent a rotation of 90^\circCCW about the origin. Each coordinate point has followed the conditions of the 90^\circCCW meaning that the point (x, y) becomes the point (-y, x).

2. For this image, look at the rotation. It is not rotated about the origin but rather about the point A. We can measure the angle of rotation:

The blue polygon is being rotated about the point A \ 145^\circ clockwise (or to the left). You would say that the blue polygon is rotated 145^\circCW 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 D. We can measure the angle of rotation:

The green polygon is being rotated about the point D \ 90^\circ counter-clockwise (or to the right). You would say that the green hexagon is rotated 90^\circCCW 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:

  1. A rotation about a point of a specific number of degrees that is positive (i.e. 90^\circ) is described as a 90^\circCCW or counter-clockwise rotation.
  2. A rotation about a point of a specific number of degrees that is negative (i.e. -90^\circ) is described as a 90^\circCW or clockwise rotation.
  3. 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 P) Rotated Image (Point P^\prime)
(0, 0) 90^\circ(or -270^\circ) (x, y) (-y, x)
(0, 0) 180^\circ(or -180^\circ) (x, y) (-x, -y)
(0, 0) 270^\circ(or -90^\circ) (x, y) (y, -x)

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 180^\circCCW rotation, what would be the coordinates of the rotated points? Show these rotations on a graph.

  1. (3, 1)
  2. (4, –2)
  3. (–5, 3)
  4. (–6, 4)
  5. (–3, –3)

If the following points were rotated about the origin with a 90^\circCW rotation, what would be the coordinates of the rotated points? Show these rotations on a graph.

  1. (–4, 3)
  2. (5, –4)
  3. (–5, –4)
  4. (3, 3)
  5. (–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.

  1. A rotation about a point of a specific degrees that is positive (i.e. 90^\circ) is described as a 90^\circCCW or counter-clockwise rotation.
  2. A rotation about a point of a specific degrees that is negative (i.e. -90^\circ) is described as a 90^\circCW or clockwise rotation.
  3. 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 P) Rotated Image (Point P^\prime)
(0, 0) 90^\circ(or -270^\circ) (x, y) (-y, x)
(0, 0) 180^\circ(or -180^\circ) (x, y) (-x, -y)
(0, 0) 270^\circ(or -90^\circ) (x, y) (y, -x)

Guidance

Quadrilateral WXYZ has coordinates W(-5, 5), X(-2, 0), Y(2, 3) and Z(-1, 3). Draw the quadrilateral on the Cartesian plane. Rotate the image 110^\circ clockwise about the point X. Show the resulting image.

Examples

Example A

Line \overline{AB} drawn from (–4, 2) to (3, 2) has been rotated about the origin at an angle of 90^\circCW. Draw the preimage and image and properly label each.

Example B

The diamond ABCD is rotated 145^\circCCW about the origin to form the image A^\prime B^\prime C^\prime D^\prime. 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 200^\circCW 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 160^\circ.

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 \overline{ST} drawn from (–3, 4) to (–3, 8) has been rotated 60^\circCW about the point S. Draw the preimage and image and properly label each.

2. The polygon below has been rotated 155^\circCCW about the origin. Draw the rotated image and properly label each.

3. The purple pentagon is rotated about the point A \ 225^\circ. 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 60^\circCW or -60^\circ.

2.

Notice the direction of the angle is counter-clockwise, therefore the angle measure is 155^\circCCW or 155^\circ.

3.

The measure of \angle BAB^\prime = m \angle BAE^\prime + m \angle E^\prime AB^\prime. Therefore \angle BAB^\prime = 111.80^\circ + 113.20^\circ or 225^\circ. Notice the direction of the angle is counter-clockwise, therefore the angle measure is 225^\circCCW or 225^\circ.

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:

  1. A rotation about a point of a specific number of degrees that is positive (i.e. 90^\circ) is described as a 90^\circCCW or counter-clockwise rotation.
  2. A rotation about a point of a specific number of degrees that is negative (i.e. -90^\circ) is described as a 90^\circCW or clockwise rotation.
  3. 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 P) Rotated Image (Point P^\prime)
(0, 0) 90^\circ(or -270^\circ) (x, y) (-y, x)
(0, 0) 180^\circ(or -180^\circ) (x, y) (-x, -y)
(0, 0) 270^\circ(or -90^\circ) (x, y) (y, -x)

Problem Set

  1. Rotate figure MATH on the grid about the origin 180^\circ.
  2. Rotate figure DEFGH about the origin -270^\circ.
  3. Rotate figure LMO about the point M by -45^\circ.
  4. Rotate figure WXY about the point W by -45^\circ.
  5. Rotate the purple figure about the origin by -90^\circ.

For each of the following diagrams rotate the images about the origin by 90^\circCCW. Find the coordinates of the preimages. On each diagram, draw and label the reflected image.

For each of the following diagrams rotate the images 60^\circCW 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 P) Rotated Image (Point P^\prime) Notation (Point P^\prime)
(0, 0) 90^\circ(or -270^\circ) (x, y) (-y, x) (x, y) \rightarrow (-y, x)
(0, 0) 180^\circ(or -180^\circ) (x, y) (-x, -y) (x, y) \rightarrow (-x, -y)
(0, 0) 270^\circ(or -90^\circ) (x, y) (y, -x) (x, y) \rightarrow (y, -x)

If, for instance you had the image below:

You would notice that the preimage is rotated in the origin 90^\circCCW. If you were to describe the rotated image using notation, you would write the following:

(x,y) \rightarrow (-y,x) \ \text{or} \ R_{90^\circ}(x,y)=(-y,x)

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 R_{90^\circ}(x,y)=(-y,x).

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 90^\circ and the direction is clockwise. Therefore the Image A has been rotated -90^\circ to form Image B. To write a rule for this rotation you would write: R_{270^\circ}(x,y)=(y,-x).

Examples

Example A

Find an image of the point (3, 2) that has undergone a rotation in:

a) about the origin at 90^\circ,

b) about the origin at 180^\circ, and

c) about the origin at 270^\circ.

Write the notation to describe the rotation.

a) Rotation about the origin at 90^\circ : R_{90^\circ}(x,y)=(-y,x)

b) Rotation about the origin at 180^\circ : R_{180^\circ}(x,y)=(-x,-y)

c) Rotation about the origin at 270^\circ : R_{270^\circ}(x,y)=(y,-x)

Example B

Rotate Image A in the diagram below:

d) about the origin at 90^\circ, and label it B.

e) about the origin at 180^\circ, and label it O.

f) about the origin at 270^\circ, and label it Z.

Write notation for each to indicate the type of rotation.

a) Rotation about the origin at 90^\circ: R_{90^\circ}A \rightarrow B=R_{90^\circ}(x,y) \rightarrow (-y,x)

b) Rotation about the origin at 180^\circ: R_{180^\circ}A \rightarrow O=R_{180^\circ}(x,y) \rightarrow (-x,-y)

c) Rotation about the origin at 270^\circ: R_{270^\circ}A \rightarrow Z=R_{270^\circ}(x,y) \rightarrow (y,-x)

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.

E: (-1, 2) \quad E^\prime: (1, -2)

Notice how both the x- and y-coordinates are multiplied by –1. This indicates that the preimage A is rotated about the origin by 180^\circCCW to form the rotated image J. Therefore the notation is R_{180^\circ}A \rightarrow J=R_{180^\circ}(x,y) \rightarrow (-x,-y).

Vocabulary

Notation Rule
A notation rule has the following form R_{180^\circ}A \rightarrow O=R_{180^\circ}(x,y) \rightarrow (-x,-y) and tells you that the image A has been rotated about the origin and both the x- and y-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 J moving from J(-2, 6) to J^\prime(6, 2). 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. J: (-2, 6) \quad J^\prime: (6, 2)

Since the x-coordinate is multiplied by –1, the y-coordinate remains the same, and finally the x- and y-coordinates change places, this is a rotation about the origin by 270^\circ or -90^\circ. The notation is: R_{270^\circ}J \rightarrow J^\prime=R_{270^\circ}(x,y) \rightarrow (y,-x)

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 E is shown in the diagram:

E(-1,3) \rightarrow E^\prime(-3,-1)

Since both x- and y-coordinates are reversed places and the y-coordinate has been multiplied by –1, the rotation is about the origin 90^\circ. The notation for this rotation would be: R_{90^\circ}(x,y) \rightarrow (-y,x).

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 C is shown in the diagram:

C(7,0) \rightarrow C^\prime(0,-7)

Since the x-coordinates only are multiplied by –1, and then x- and y-coordinates change places, the transformation is a rotation is about the origin by 270^\circ. The notation for this rotation would be: R_{270^\circ}(x,y) \rightarrow (y,-x).

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) 90^\circ (or -270^\circ) R_{90^\circ}(x,y)=(-y,x)
(0, 0) 180^\circ (or -180^\circ) R_{180^\circ}(x,y) \rightarrow (-x,-y)
(0, 0) 270^\circ (or -90^\circ) R_{270^\circ}(x,y) \rightarrow (y,-x)

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 90^\circ Rotation 180^\circ Rotation 270^\circ Rotation 360^\circ 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 B is between A and C, then B^\prime will be between A^\prime and C^\prime.

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 ABC is drawn such that the vertices are at A(1, 1), B(5, 5) and C(-2, 4). Triangle A^\prime B^\prime C^\prime is rotated 45^\circCW.

a) What are the coordinates of Triangle A^\prime B^\prime C^\prime ?

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 A^\prime B^\prime C^\prime is a rotation of triangle ABC?

a) The coordinates of A^\prime B^\prime C^\prime upon rotation 45^\circCW would be:

Preimage ABC A(1, 1) B(5, 5) C(-2, 4)

Reflected Image A^\prime B^\prime C^\prime A^\prime (-0.7, 5) B^\prime (5, 5) C^\prime (-0.7, -9.2)

b) The measures of each angle are shown in the diagram.

Congruent angles have the same measure. Since \angle A= \angle A^\prime, \angle B=\angle B^\prime, and \angle C= \angle C^\prime, the angles are congruent.

c)

The side lengths are also the same (m \overline{AC}=m \overline{A^\prime C^\prime},m \overline{BC}=m \overline{B^\prime C^\prime}, and m \overline{AB}=m \overline{A^\prime C^\prime}).

d) Since the angle measures are the same and the lengths of each side are the same, the triangles are congruent. Triangle ABC has been reflected to produce triangle A^\prime B^\prime C^\prime.

Examples

Example A

Graph the following two squares to determine if Quadrilateral A is rotated to form Quadrilateral B.

Quadrilateral A Vertices: W(2, -2), X(5, 0), Y(5, 3), and Z(4, 3).

Quadrilateral B Vertices: W^\prime (2, 2), X^\prime (0, 5), Y^\prime (-3, 5), and Z^\prime (-3, 4).

The side lengths are the same (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}, and m \overline{XY}=m \overline{X^\prime Y^\prime}) and the angle lengths are the same (\angle X= \angle X^\prime, \angle Y=\angle Y^\prime, \angle W= \angle W^\prime, and \angle Z= \angle Z^\prime).

Since the side and angle measures are the same Quadrilateral A is congruent to Quadrilateral B. Each y-coordinate on ABCD 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 90^\circCCW.

Example B

Describe how you would know if image T is rotated onto image T^\prime.

As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image T is congruent to Image T^\prime. Each x- and y-coordinate is multiplied by –1. With all of this, you can conclude that the Image T is rotated about the origin 180^\circ to form Image T^\prime.

Example C

Describe how you would know if image R is rotated to form image R^\prime.

As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image R is congruent to Image R^\prime. Looking at point E and E^\prime, the x- and y-coordinates have been reversed and then the x-coordinate has been multiplied by –1. With all of this, you can conclude that Image R is rotated across the origin at 90^\circCW to form the image R^\prime.

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 x-coordinate point is multiplied by –1 while each y-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 90^\circCW 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 x- and y-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 180^\circ 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 J^\prime. Each y-coordinate has been multiplied by –1. With all of this, you can conclude that Image J^\prime is a rotated image of the Preimage J. As well, Preimage J has been rotated across the origin at 270^\circCCW or 90^\circCW in order to produce Image J^\prime.

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.

  1. Triangle ABC with vertices A(3, 1), B(0, -1) and C(1, 5) has been rotated about the point P(-1, 1) \ 180^\circ to produce triangle A^\prime B^\prime C^\prime. Draw the diagram representing this rotation and prove the image A^\prime B^\prime C^\prime is actually a rotation.
  2. Triangle DEF with vertices D(0, 1), E(5, 1) and F(2, 5) has been rotated about the point P(-2, 5) \ 90^\circCCW to produce triangle D^\prime E^\prime F^\prime. Draw the diagram representing this rotation and prove the image D^\prime E^\prime F^\prime is actually a rotation.
  3. Draw a letter from the alphabet that produces the same letter after a 180^\circ rotation. Prove the rotated image is actually a rotation.
  4. Draw another letter from the alphabet that produces the same letter after a 180^\circ rotation. Prove the rotated image is actually a rotation.
  5. 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.

  1. A rotation about a point of a specific number of degrees that is positive (i.e. 90) is described as a 90^\circCCW or counter-clockwise rotation.
  2. A rotation about a point of a specific number of degrees that is negative (i.e. -90^\circ) is described as a 90^\circCW or clockwise rotation.
  3. 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 P) Rotated Image (Point P^\prime)
(0, 0) 90^\circ (or -270^\circ) (x, y) (-y, x)
(0, 0) 180^\circ (or -180^\circ) (x, y) (-x, -y)
(0, 0) 270^\circ (or -90^\circ) (x, y) (y, -x)

Therefore if you were asked to rotate an image about the origin at 90^\circCCW, you quickly knew to multiply the y-coordinate by –1 and then to interchange the x- and y-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) 90^\circ (or -270^\circ) R_{90^\circ}(x,y)=(-y,x)
(0, 0) 180^\circ (or -180^\circ) R_{180^\circ}(x,y) \rightarrow (-x,-y)
(0, 0) 270^\circ (or -90^\circ) R_{270^\circ}(x,y) \rightarrow (y,-x)

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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May 28, 2014

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Jan 14, 2015
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