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5. Rigid Transformations
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Geometry

5.05 Sequences of transformations

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Sequences of transformations

A sequence 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 \left(x,y\right) \to \left(x+4, y\right)

  • Rotate 90\degree clockwise about the origin \left(x,y\right) \to \left(y, -x\right)

  • Reflect across the y-axis \left(x,y\right) \to \left(-x, y\right)

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.

Examples

Example 1

Given the following transformations to the pre-image \left(5,-1\right), identify the coordinates of the image:

  • Rotated 90 \degree clockwise about the origin;

  • Translated 3 units up, and 2 units left;

  • Reflected across the y-axis.

Worked Solution
Create a strategy

We need to do the transformations in the order they are given.

  • Rotating the point 90 \degree clockwise is the same as rotating the point 270 \degree counterclockwise. The mapping for rotating something counterclockwise 270 \degree about the origin is \left(x,y \right) \to \left(y,-x\right).

  • Translating the point 3 units up and 2 units left has the mapping \left(x,y \right) \to \left(x-2,y+3\right)

  • Reflection across the y-axis has the mapping \left(x,y \right) \to \left(-x,y\right).

Apply the idea
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First, we rotate the point \left(5,-1\right) clockwise 90\degree about the origin to get the point \left(-1,-5\right).

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Next, we translate the point \left(-1,-5\right) up three units and left two units to get \left(-3,-2\right).

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Finally, we reflect the point \left(-3,-2\right) across the y-axis to get \left(3,-2\right).

Example 2

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

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Worked Solution
Create a strategy

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.

Apply the idea

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. The transformation mapping is \left(x, y\right) \to \left(x-1, y+5\right).

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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.

Reflect and check

There is more than one way to obtain an image from a pre-image using a sequences 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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Example 3

The vertices of triangle ABC have the coordinates A\left(-2, 4\right), B\left(-1,3\right), and C\left(-3,2\right). The following sequence of transformations is performed:

  • Rotation by 180 \degree clockwise about the origin, then

  • Reflection across the x-axis.

What equivalent single transformation will take triangle ABC to triangle A''B''C''?

Worked Solution
Create a strategy

We can perform the rotation to get triangle A'B'C', then the reflection to get triangle A''B''C''. Then we can compare the location and orientation of the triangles.

Apply the idea
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We can see that the vertices of triangle A''B''C'' have coordinates A''\left(2,4\right), B''\left(1,3\right), and C''\left(3,2\right). Comparing to the vertices of triangle ABC, only the sign of the x-coordinates have changed, so the single transformation from ABC to A''B''C'' is a reflection across the y-axis.

Reflect and check

A rotation by 180 \degree clockwise about the origin has the transformation mapping: \left(x, y\right) \to \left(-x, -y\right)

A reflection across the x-axis has the transformation mapping: \left(x,y\right) \to \left(x, -y\right)

The angle measures and the side lengths of the pre-image are preserved when comparing to its image.

Idea summary

We can perform or check a sequence of transformations by working with one transformation at a time.

Outcomes

G.RLT.3

The student will solve problems, including contextual problems, involving symmetry and transformation.

G.RLT.3ci

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: i) translations;

G.RLT.3cii

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: ii) reflections over any horizontal or vertical line or the lines y = x or y = -x;

G.RLT.3ciii

Given an image or preimage, identify the transformation or combination of transformations that has/have occurred. Transformations include: iii) clockwise or counterclockwise rotations of 90°, 180°, 270°, or 360° on a coordinate grid where the center of rotation is limited to the origin;

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