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4. Rigid Motions
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United States of America
Grades 9-12

4.03 Rotations

Lesson
Practice

Introduction

Rotations are the final rigid transformation introduced in 8th grade. We will now learn about new notation for the rotation of figures on the coordinate plane, perform rotations, and verify rotations.

Rotations

Exploration

Move the slider to create a rotating image of the figure and check the boxes to explore.

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  1. What do you notice about the distance segments that are the same color?
  2. What do you notice about the angles formed by the pairs of same colored segments?

A rotation is a transformation of a figure by turning it about a point called the point of rotation. The amount of rotation can be expressed in the number of degrees. The direction of the rotation for two-dimensional figures can be described in the clockwise or counterclockwise direction. When it is not specified we assume the direction is counterclockwise.

Connecting the center of rotation to any point on the pre-image and to its corresponding point on the rotated image, the line segments are equal in length and the measure of the angle formed is the angle of rotation.

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A rotation can be denoted in function notation in the form R_{\theta\degree,P}(\text{Name of shape}) where \theta\degree is the measure of the directed angle (the angle of rotation in the counterclockwise direction) and P is the point of rotation.

When rotated about the origin, the transformation mappings are as follows:

  • Degree of rotation counterclockwise: 90 \degree \, \, \, \qquad Transformation mapping: \left(x,y \right) \to \left(-y,x\right)
  • Degree of rotation counterclockwise: 180 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(-x,-y\right)
  • Degree of rotation counterclockwise: 270 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(y,-x\right)
  • Degree of rotation counterclockwise: 360 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(x,y\right)

If the point of rotation is not at the origin we will not have such nice transformation maps, but the principles of rotation remain the same.

Examples

Example 1

\overline{AB} has been rotated counterclockwise about the origin.

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a

Describe the rotation required to map \overline{AB} to \overline{A'B'}.

Worked Solution
Create a strategy

We can identify the rotation that has taken place by drawing connecting segments from the point of origin, O, to point B and to point B', and then measuring the angle.

Apply the idea

In this case, \angle BOB' creates a straight line. The segment has been rotated clockwise 180\degree.

b

Write the transformation mapping.

Worked Solution
Create a strategy

We know the function notation mapping is R_{\theta\degree,P}(\text{Name of shape}) where \theta\degree is the angle of rotation in the counterclockwise direction, and P is the point of rotation.

Apply the idea

Function notation: R_{180\degree,O}(\overline{AB})=\overline{A'B'}

We know that the point of rotation is the origin, O\left(0,0\right).

We can see that A\to A' maps \left(3, 3\right) \to \left(-3,-3\right) and B\to B' maps \left(2, 1\right) \to \left(-2,-1\right). Both are of the form\left(x,y\right)\to\left(-x,-y\right) which corresponds to a 180\degree counterclockwise rotation about the origin.

Example 2

Consider the figure ABCD:

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a

Determine the image of ABCD when rotated about the point A by 180 \degree.

Worked Solution
Create a strategy

We can determine the image of each vertex rotated 180\degree about A by considering their translation from A and then rotating that translation.

Consider that, when rotating 180 \degree,:

  • A point to the right of center ends up left of center
  • A point below center ends up above center
  • A point to the left of center ends up right of center
  • A point above center ends up below center
Apply the idea

In the pre-image, we can get to the other vertices from A by the following translations:

  • A\to B: translate 2 units to the right
  • A\to C: translate 2 units to the right and 2 units up
  • A\to D: translate 2 units up

If we apply a rotation of 180\degreeto these translations, we can get the images of the vertices:

  • A\to B': translate 2 units to the left
  • A\to C': translate 2 units to the left and 2 units down
  • A\to D': translate 2 units down

We can plot this as:

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Reflect and check

Note that a rotation of 180 \degree clockwise or counterclockwise will end in the same location, so we do not have to specify direction in this case.

b

Use the points D and D' to verify the rotation.

Worked Solution
Create a strategy

By connecting the center of rotation, A, to points D and D', verify that the line segments are equal in length and the measure of the angle of rotation.

Apply the idea

Using the coordinate grid with the pre-image and image from part (a), the line segment from the point of rotation A to D is 2 units.

The line segment from the point of rotation A to D' is also 2 units.

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We can see that the measure of the angle of rotation between the corresponding points is 180 \degree, which is the rotation instruction from part (a).

The length of the line segments from the point of rotation to each of two corresponding points are equivalent and the measure of the angle of rotation is the same as the rotation of the figure.

Reflect and check

We can also use tracing paper and a protractor to verify that the figure was rotated 180 \degree.

c

Determine the image of ABCD when rotated about the point \left(2,0\right) by 90\degree clockwise.

Worked Solution
Create a strategy

A rotation of 90\degree clockwise will have the following effect on translations:

  • Left \to Up
  • Up \to Right
  • Right \to Down
  • Down \to Left
Apply the idea

Using the same approach as for part (a), we can note that:

  • \left(2,0\right)\to A: translate 1 unit left and 1 unit down
  • \left(2,0\right)\to B: translate 1 unit right and 1 unit down
  • \left(2,0\right)\to C: translate 1 unit right and 1 unit up
  • \left(2,0\right)\to D: translate 1 unit left and 1 unit up

If we apply a rotation of 90\degree clockwise to these translations, we can get the images of the vertices:

  • \left(2,0\right)\to A': translate 1 unit up and 1 unit left
  • \left(2,0\right)\to B': translate 1 unit down and 1 unit left
  • \left(2,0\right)\to C': translate 1 unit down and 1 unit right
  • \left(2,0\right)\to D': translate 1 unit up and 1 unit right

Using these translations, we can see that

  • A'=D
  • B'=A
  • C'=B
  • D'=C

In other words, the rotation maps ABCD onto itself.

We can plot this as:

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Example 3

Sketch the result of the rotation R_{135\degree,B}\left(ABCD\right), using the figure ABCD that is shown.

Quadrilateral A B C D.
Worked Solution
Create a strategy

The rotation function tells us that:

  • the directed angle is 135\degree, so we want to rotate the figure 135\degree counterclockwise
  • the point of rotation is B
  • the object being rotated is the quadrilateral ABCD

Combining this information, we want to rotate the quadrilateral ABCD about its vertex B by 135\degree counterclockwise.

Since the figure is not on a coordinate plane, we will need a protractor and ruler to perform the rotation.

Apply the idea

Measure each side length, set the protractor to 135 \degree, then draw each segment at the directed angle with B at the center of rotation. Check the length of the segments of the image using a ruler.

Quadrilateral A B C D sharing vertex B with quadrilateral A prime B C prime D prime.
Reflect and check

Connecting the center of rotation to any point on the pre-image and to its corresponding point on the rotated image, we can use our ruler to confirm the line segments are equal in length and our protractor to confirm that the measure of the angle formed is the angle of rotation. The following angles should each measure 135 \degree: \angle ABA', \angle CBC', and \angle DBD'.

Idea summary

When rotated counterclockwise about the origin, the transformation mappings are as follows:

  • Degree of rotation: 90 \degree \, \, \, \qquad Transformation mapping: \left(x,y \right) \to \left(-y,x\right)
  • Degree of rotation: 180 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(-x,-y\right)
  • Degree of rotation: 270 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(y,-x\right)
  • Degree of rotation: 360 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(x,y\right)

If the point of rotation is not at the origin we will not have such nice transformation maps, but the principles of rotation remain the same.

Outcomes

G.CO.A.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

G.CO.A.4

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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