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3. Rigid Motions
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Geometry

3.03 Rotations

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Concept summary

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.

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A rotation can be denoted in functional notation in the form R_{\theta\degree,P}(\text{Shape being rotated}) 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. For any point being rotated, the function representation R_{\theta\degree,P}\left(A\right)=A\rq means that m\angle APA\rq=\theta\degree, and also that AP=A\rq P.

If the input for a rotation is a figure, then we will have that any corresponding points, A and A\rq, in the pre-image and image will have these relationships.

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.

If a rotation maps a figure onto itself, then we say that the figure has rotational symmetry about that point. We can call that point the shape's center of rotation.

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For example, if we rotate the square about the origin by 90\degree counterclockwise, we can check that each point on the square will be mapped to another point on the square.

The result is that the image of the rotation will be the same as the pre-image.

We can see that the square also has rotational symmetry about its center for rotations of 180\degree and 270\degree as well.

Worked 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'}.

Approach

We can identify the rotation that has taken place by looking at the coordinates of A and B in the pre-image and comparing them to the coordinates of A' and B' in the image.

Solution

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 rotation about the origin.

Reflection

We can see that the entire shape has moved from the first quadrant to the third quadrant which, when rotating about the origin, will always coincide with a 180\degree rotation in either clockwise or counterclockwise direction.

b

Write the transformation mapping.

Approach

We know the functional notation mapping is R_{\theta\degree,P}(\text{Shape being rotated}) where \theta\degree is the angle of rotation in the counterclockwise direction, and P is the point of rotation.

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

We also found in part (a) A\to A' and B\to B' followed the mapping \left(x,y\right)\to\left(-x,-y\right) which corresponds to a 180\degree rotation about the origin.

Solution

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

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.

Approach

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

Consider that, when rotating 180\degree, we have that:

  • Right \to Left
  • Down \to Up
  • Left \to Right
  • Up \to Down

Solution

In the pre-image, we can get to the other corners 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 corners:

  • 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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b

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

Approach

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

Solution

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 corners:

  • \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.

Example 3

Identify the rotation(s) that map ABCD onto itself.

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Approach

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If we look at the point of intersection of the axes of symmetry we can find the point of rotation for rotational symmetry.

Solution

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Rotating about the point \left(1,2\right) by 90 \degree clockwise or 270 \degree counterclockwise, we can see that we do not get rotational symmetry.

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Rotating about the point \left(1,2\right) by 180 \degree clockwise or counterclockwise, we can see that we get rotational symmetry.

Rotating about the point \left(1,2\right), we can see that we only get rotational symmetry when we rotate by 180\degree clockwise about the point \left(1,2\right).

This is the function R_{180\degree,\left(1,2\right)}

Example 4

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

Approach

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.

Solution

Reflection

To find the rotation of the whole figure, it is enough to find the rotation of the vertices of ABCD, then connect them to complete the image of the rotation.

Since B is the point of rotation, its image will be itself. The other three vertices will have images which form directed angles of 135\degree with their pre-images and the point of rotation. As we can see:m\angle ABA\rq = m\angle CBC\rq = m\angle DBD\rq = 135\degree

Outcomes

G.CO.A.1

Describe transformations as functions that take points in the plane (pre-image) as inputs and give other points (image) as outputs. Compare transformations that preserve distance and angle measure to those that do not, by hand for basic transformations and using technology for more complex cases.

G.CO.A.2

Given a rectangle, parallelogram, trapezoid, or regular polygon, determine the transformations that carry the shape onto itself and describe them in terms of the symmetry of the figure.

G.CO.A.3

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

G.CO.A.4

Given a geometric figure, draw the image of the figure after a sequence of one or more rigid motions, by hand and using technology. Identify a sequence of rigid motions that will carry a given figure onto another.

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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