Topics
4. Rigid Motions
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United States of America
Grades 9-12

4.01 Translations

Lesson
Practice

Introduction

We saw how to use translations to move figures on the coordinate plane in 8th grade, looking for patterns and performing translations. We will continue to perform translations here, and apply new notation to describe transformations.

Translations

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 are the points that 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. The points that make up the image would be considered the outputs of the function transformation.

Exploration

Drag the points on the object to change its shape. Then, drag the sliders to create its image after a translation.

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  1. If you are given the coordinates of the image, what information would you need to find the coordinates of the pre-image and vice versa?
  2. What can we conclude about the corresponding segment between any two points on the pre-image and its image?

When performing a translation, connecting any point on the pre-image to its corresponding point on the translated image, and connecting a second point on the pre-image to its corresponding point on the translated image, the two segments are equal in length, translate in the same direction, and are parallel.

It is common to label the vertices of figures with letters and to use an apostrophe, called a prime, to label vertices 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 horizontally, and k units vertically to obtain the image.

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

The movement h units horizontal and k units vertical 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.

An image of a polygon and its pre image. Rays are drawn from each vertex of the pre image to its correspnding image point. The rays are parallel.

Examples

Example 1

For the following graph:

a

Describe the transformation in words.

Worked Solution
Create a strategy

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

Apply the idea

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

Reflect and check

It can be easier to work out the description by looking at one vertex and seeing how that moved. Then we can confirm by trying the same translation with the other vertices, or even other points on the sides of the triangle.

b

Write the transformation in function notation.

Worked Solution
Create a strategy

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.

Apply the idea

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

c

Use the definition of a translation to verify that the transformation is a translation.

Worked Solution
Create a strategy

Recall that, when a translation has occurred, any segments created by connecting points on the pre-image to their corresponding points on the image will be equal in length, parallel, and translate the points in the same direction.

Apply the idea

Based on the description and function notation in parts (a) and (b), we can confirm that every point on the pre-image is moved the same distance and in the same direction to lead to the image.

We can connect each vertex on the pre-image to its corresponding vertex on its image and find their slopes and lengths.

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We can find the slope of each segment by counting the vertical rise and horizontal run from one endpoint to the other. For the segment connecting the upper vertex of each triangle the {\text{Rise } = -2} and the \text{Run }=-4 so the slope is \dfrac{-2}{-4}=\dfrac{1}{2}. Repeating this process for the other segments we will find that they each have a slope of \dfrac{1}{2}. Since all segments have the same slope, they are parallel.

To calculate the length of each segment we can use the slope triangles we have drawn and the Pythagorean theorem.

\displaystyle a^2+b^2\displaystyle =\displaystyle c^2Pythagorean theorem
\displaystyle 2^2+4^2\displaystyle =\displaystyle c^2Substitute the side lengths of slope triangle
\displaystyle 4+16\displaystyle =\displaystyle c^2Evaluate the exponents
\displaystyle 20\displaystyle =\displaystyle c^2Evaluate the addition
\displaystyle \sqrt{20}\displaystyle =\displaystyle cTake the square root of both sides

By repeating the same process for the other two directed segments we will find that they each have a length of \sqrt{20}.

Since the length of the segments from the vertices on the pre-image to its image are the same length and parallel to each other meaning every point on the pre-image is moved the same distance and in the same direction, the transformation is a translation.

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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Worked Solution
Create a strategy

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.

Apply the idea

Point \left(1,3 \right) becomes \left(1+1,3-4\right) or \left(2,-1\right).

Point \left(2,1 \right) becomes \left(2+1,1-4\right) or \left(3,-3\right).

Point \left(3,2 \right) becomes \left(3+1,2-4\right) or \left(4,-2\right).

Point \left(2,3 \right) becomes \left(2+1,3-4\right) or \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.

Worked Solution
Create a strategy

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).

Apply the idea

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-\left(-2\right)=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, 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.

Worked Solution
Create a strategy

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.

Apply the idea
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Reflect and check

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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Idea summary

Since a translation is a transformation in which every point in a figure it moved in the same direction and by the same distance, any two corresponding segments on a pre-image and its image will be equal in length, translate in the same direction, and are parallel.

Outcomes

G.CO.A.2

Represent transformations in the plane using, e.g. Transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. Translation versus horizontal stretch).

G.CO.A.4

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

G.CO.A.5

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g. Graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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