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10. Connecting Algebra & Geometry with Coordinates
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Integrated Math 1

10.01 Rigid transformations

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Concept summary

A rigid motion (or rigid transformation) is a transformation that preserves distances and angle measures.

A transformation of a figure is a mapping that changes the figure's size or position in space, including rotation. We can also think of a transformation as a function, where the input values make up the figure that is being transformed.

The figure before it is transformed is called the pre-image. The figure after it has been transformed is called the image.

It is common to label the corners of figures with letters and to use a dash, called a prime, to label corners of the transformed image. For example, if A was the pre-image, then A' (spoken as "A-prime") is the image.

Translation

A transformation in which every point in a figure is moved in the same direction and by the same distance.

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We have two ways to describe a translation algebraically:

  • Coordinate form: The translation \left(x,y\right) \to \left(x+h,y+k\right) takes the pre-image and moves it h units to the right, and k units up to obtain the image.

  • Function notation: The translation T_{<h,k>}(A) takes the pre-image, A, and moves it h units to the right and k units up.

A reflection across the line of reflection is a transformation that produces the mirror image of a geometric figure.

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Line of reflection

A line that a figure is flipped over to create a mirror image

The main lines of reflections have the following impact on a point:

  • Line of reflection: x-axis \qquad Transformation mapping: \left(x, y\right) \to \left(x, -y \right)

  • Line of reflection: y-axis \qquad Transformation mapping: \left(x, y\right) \to \left(-x, y \right)

  • Line of reflection: y=x \, \, \qquad Transformation mapping: \left(x, y\right) \to \left(y, x \right)

  • Line of reflection: y=-x \, \, \, \quad Transformation mapping: \left(x, y\right) \to \left(-y, -x \right)

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 degress. 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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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)

A composition of transformations is a list of transformations that are performed one after the other. These transformations include translations, rotations, and reflections.

When performing multiple transformations one after the other, the pre-image for each new transformation will be the image of the previous transformation.

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For example, suppose that we applied the following transformations to the point \left(-1,2\right):

  • Translate 4 units to the right
  • Rotate 90\degree clockwise about the origin
  • Reflect across the y-axis

We perform the translation first, then rotate the image of the translation, then reflect the image of the rotation.

When performing multiple transformations, the order in which they are applied matters.

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Suppose that we applied the following transformations to the point \left(-1,2\right):

  • Reflect across the y-axis
  • Rotate 90\degree clockwise about the origin
  • Translate 4 units to the right

As we can see, performing the transformations in a different order results in a different image of the transformations.

Worked examples

Example 1

For the following graph:

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a

Describe the translation in words.

Approach

We need to identify the direction the pre-image has been moved and the distance each point has moved to obtain the image.

Solution

The pre-image has been moved four units to the left and two units down to obtain the image.

Reflection

It can be easier to work out the description by looking at one corner and seeing how that moved.

b

Write the translation in function notation.

Approach

We know from part (a) that the pre-image has been moved four units to the left and two units down to obtain the image.

Solution

T_{\langle -4,-2 \rangle}\left(\triangle ABC\right) = \triangle A'B'C'

Example 2

For the following graph:

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a

Identify the line of reflection.

Approach

The line must be equal distance between the pairs A and A', B and B', and C and C'. So to find the line of reflection we find the average between each pair and then find the line that goes through all the averages.

Solution

The average of A and A': \left(7,0\right)

The average of B and B': \left(-8,0\right)

The average of C and C': \left(8,0\right)

The line of reflection goes through all of the above points. Therefore the line of reflection is the x-axis, y=0.

b

Write the transformation mapping in both coordinate and function notation.

Approach

The reflection mapping is given either by the change in arbitrary point \left(x,y\right), or by using function notation R_{\text{line of reflection}}(\text{shape})

Solution

Coordinate notation: (x,y) \to (-x,y)

Function notation:R_{y=0}(\triangle ABC) = \triangle A'B'C'

Reflection

There are multiple ways of representing the same transformation.

Example 3

\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 preimage 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 4

Describe the transformations required to obtain the image from the pre-image.

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Approach

First, we need to look to see if the side lengths are in the same order when looking at them clockwise, or in the reverse order. If the side lengths are in the same order when looking at them clockwise, then the pre-image is rotated. If they are in the reverse order, then the pre-image has been reflected.

Then, we need to see if it needs to be moved up, down, left or right.

Solution

The side lengths are in the reverse order, so a reflection is required. So, lets reflect the pre-image across the line y = -x.

Remember that when the line of reflection is y=-x, the transformation mapping is \left(x, y\right) \to \left(-y, -x \right)

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Now, the pre-image needs to be moved one unit to the left, and five units up.

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The transformations to the pre-image to get the image are:

  • Reflection across the line y = -x.
  • Translate 1 unit left, and 5 units up.

Reflection

There is more than one way to obtain an image from a pre-image using a composition of functions.

Another solution is:

  1. Reflect across the line x=3

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  2. Rotate 90\degree counterclockwise about the point \left(3,1\right).

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  3. Translate 5 units left and 1 unit up.

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Outcomes

M1.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.

M1.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.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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