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6. Rigid Motions
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United States of AmericaTennessee
Integrated Math 2

6.01 Translations

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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.

The movement h units right and k units up can be represented as a directed line segment. So we can think of a translation as moving the pre-image along the directed line segment to get the image.

Since every point in the pre-image is moved in the same direction and distance, every line segment from a pre-image point to its corresponding image point will be parallel to the directed line segment that represents the translation.

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

Draw the image given from the transformation \left(x,y\right) \to \left(x+1,y-4\right) on the pre-image:

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Approach

The transformation tells us each value of x needs to be increased by 1 and each value of y needs to be decreased by 4. We can take each point on the figure and do these transformations to them.

Solution

Point \left(1,3 \right) when tranformed becomes point \left(2,-1\right).

Point \left(2,1 \right) when tranformed becomes point \left(3,-3\right).

Point \left(3,2 \right) when tranformed becomes point \left(4,-2\right).

Point \left(2,3 \right) when tranformed becomes point \left(3,-1\right).

Plotting these points on the graph, we obtain the image.

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Example 3

Consider the figure \triangle ABC and the directed line segment v.

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a

Describe the translation represented by the directed line segment v.

Approach

We can identify the horizontal and vertical movement of the translation by considering the change in x- and y-values from the start to the end of the directed line segment.

For the directed line segment v, we can see that that starting point is \left(-2,4\right) and the ending point is \left(3,1\right).

Solution

Since the x-value for the directed line segment changes from x=-2 to x=3, the horizontal movement of the translation will be 3-(-2)=5 units to the right.

Since the y-value for the directed line segment changes from y=4 to y=1, the vertical movement of the translation will be 1-4=-3 units up, which is equivalent to 3 units down.

Altogether, we can describe the translation represented by the directed line segment v to be 5 units right and 3 units down.

b

Translate the figure \triangle ABC by the directed line segment v.

Approach

We want to translate the whole triangle by the directed line segment. We can do this by translating just the vertices, then joining them up to form the full image.

To translate each vertex, we can apply the movements described in the previous part.

Solution

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Reflection

If we construct directed line segments from each vertex in the pre-image to its corresponding vertex in the image, we can see that they are all parallel to the directed line segment v, and are also the same length.

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If we had a way to move the directed line segment without changing its direction, we could use it to easily translate points from the pre-image into the image.

Outcomes

M2.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.

M2.G.CO.A.3

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

M2.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.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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