Topics
3. Rigid Motions
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United States of AmericaFL
Grade 912

3.01 Translations

Lesson
Practice
Lesson

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.

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

Translation

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

2
4
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x
2
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y

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.

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.

Worked examples

Example 1

For the following graph:

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-2
-1
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x
-4
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-2
-1
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y
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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x
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y

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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x
-4
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y

Outcomes

MA.912.GR.2.1

Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

MA.912.GR.2.3

Identify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

MA.912.GR.2.5

Given a geometric figure and a sequence of transformations, draw the transformed figure on a coordinate plane.

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