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Congruence, Proof, and Constructions

Translations of Figures in a Plane

Besides rotations, another common type of transformation is translations. As the name suggests, the effect a translation has on an object is a slide across a plane. In this lesson, the formal definition and properties of a translation will be developed.

Catch-Up and Review

Here is some recommended reading before getting started with this lesson.

Explore

Translations Using Vectors

In the following applet, the magnitude and direction of the vector labeled can be set. Then, can be translated along the vector.
Once the translation is performed, is there any relationship between and

Discussion

Properties of Translations

From the exploration, the following conclusions can be drawn.

  • The lengths of and are equal to the magnitude of
  • The three segments and are parallel to

To perform a translation, a vector is required. This implies that the direction of the translation plays an important role. To illustrate this statement, consider the following diagram.

To triangles such that AA', BB', and CC' are all parallel and congruent

It can be seen that the lengths of and are equal to the magnitude of Also, and are parallel to Therefore, the same two conclusions written above apply to this diagram. However, is not a translation of This is because, although vectors and have the same direction as the vector does not have the same direction as

Vectors BB' and CC' have the same direction as vector v, while AA' has the opposite direction

Consequently, and referring to what can be inferred from the exploration applet, a third conclusion can be drawn.

The vectors and have the same direction as

Discussion

Definition of Translations

Be aware that the three conclusions written before do not depend on the preimage. They hold true even when the preimage is a single point, a segment, a polygon, or any other figure. Then, these properties can be used to define a translation properly.

Concept

Translation

A translation is a transformation that moves every point of a figure the same distance in the same direction. More precisely, a translation along a vector maps every point in the plane onto its image such that the following statements hold true.

Notice that these three properties imply that the quadrilateral formed by the tip of and the tail of is a parallelogram.
Vector v, point A and its image A under a translation along v
Since translations preserve side lengths and angle measures, they are rigid motions.

Explore

Translating a Figure

By definition of translation, there is a relationship between the vector that defines this transformation and the segment connecting a point with its image. Below it will be explored whether there is a relationship between a figure and its image after a translation. In the applet, the measure of and the length of its sides can be set. Also, the magnitude and direction of can be defined. Once done, can be translated along
After performing the translation, consider the following questions.
  • What do the preimage and the image have in common?
  • Apart from the dimensions, is there any other relationship between and
  • Do and have the same relationship as and

Example

Identifying Translations

As the previous exploration shows, translations preserve side lengths and angle measures. That confirms translations are rigid motions. Additionally, take note that translations map segments onto parallel segments. Consider the polygon shown in the middle of the diagram below. The other polygons are images of after different transformations.

What polygons are the image of after a translation?

Answer

Only polygons and are the image of after a translation.

Hint

Remember that translations map segments onto parallel segments. Since looks like an arrow pointing up, the images of after a translation will also look like an arrow pointing up.

Solution

Since translations map segments onto parallel segments, the sides of are parallel to the sides of its image after any translation. Notice that looks like an arrow pointing up.

Therefore, the image of after any translation will also look like an arrow pointing up. In the given diagram, it can be seen that only and satisfy this condition. Consequently, these polygons are images of after a translation.
To obtain the other polygons, must be translated and a rotated.

Example

Translations and Polygon's Vertices

In the previous example, only the shape of polygon was used to determine which polygons were the image of after a translation. When the vertices are labeled, keep an eye on them. Consider the following three squares.

Which of the squares or if either, is the image of after a translation?

Hint

Pay close attention to the labels of the vertices. Remember that, for every preimage and image must be parallel to the vector that defines the translation. This means that all the segments connecting a point to its image are parallel to each other.

Solution

At first glance, it seems like both squares and are the image of after a translation. In fact, if the vertices were not labeled, that would be the case.
However, because the vertices are labeled, closer attention needs to be paid. By definition of translation, for every preimage and its image the following relations hold true.
  • The segment whose endpoints are and has the same length as the magnitude of the vector that defines the translation. This means all the segments connecting a point to its image have the same length.
  • The segment whose endpoints are and is parallel to the vector that defines the translation. This means that all the segments connecting a point to its image are parallel to each other.
With the above information in mind, the segments that connect a vertex and its image will be drawn.
Considering only the squares and the following observations about the segments that connect the vertices and their corresponding images can be made.
  • These segment have different lengths.
  • These segments are not parallel.

Therefore, is not the image of after a translation. Conversely, satisfies the two conditions previously written. Even more, since every vector that connects a vertex to its image have the same direction, it can be concluded that is a translation of

Discussion

Constructing a Translation

Translations can be performed by hand with the help of a straightedge and a compass.

Triangle ABC and vector PQ

To translate along follow the four steps below.

1

Draw the Line Containing

Using a straightedge, draw the line that contains the vector that defines the translation. In this case, this vector is

2

Draw Parallel Lines to Through Each Vertex

Draw parallel lines to through each vertex.

3

Copy Over the Lines Drawn
Over the three lines drawn in the previous step, using a compass copy vector in such way that each vertex of is the tail of the new vector.
The tip of each vector is the image of each vertex.

4

Draw the Image

Draw the triangle formed by the image points and

Extra

Alternative Method

By definition of translation, vectors and are parallel and have the same magnitude and direction. This makes a parallelogram. With this in mind, the translation of can be performed as follows.

  • Copy the length of : open the compass to the length of Then, with this setting, place the compass tip on each vertex of and draw an arc.
  • Copy the length from to each vertex: place the compass tip on and open the compass to Then, with this setting, place the compass tip on and make an arc that intersects the arc centered at The point of intersection is
    Here, is a parallelogram. Repeating the same process, the image of the rest of the vertices can be found.
  • Draw the Image: Draw the triangle formed by the image points and

Discussion

Component Form of a Translation Vector

In the coordinate plane, the component form of the translation vector is closely related to the coordinates of the image of a point Investigate this relationship by using the following applet.
Applet to investigate the coordinates of a point after a translation along vector (v1,v2).
As it can be seen in the applet, the image of after a translation along the vector is the point In other words, to find the image of , add to the coordinate and to the coordinate. When a figure and its image after a particular translation are given, it is possible to find the translation vector. In fact, the translation vector is any vector connecting a preimage with its image. That is, any vector of the form

Pop Quiz

Practice Translations

In the following applet, one of the following tasks may be required.

  • Translate along the given vector
  • Write the component form of the translation vector that maps onto

To translate place points and where they should be after the translation is applied.

Performing random translations to random triangles

Example

Composition of Translations

When learning about rotations, it was said that the composition of two rotations could be a translation. Now, the composition of two translations will be examined.

Consider the following pair of quadrilaterals and Also, consider a pair of different translations. One translation along vector and the other along

Pair of quadrilaterals in the coordinate plane
a In which order must the translations be applied to so that it is mapped onto Only one option is correct.
b Is there a single translation that maps onto
If so, write the component form of the translation vector.

Hint

a Translating along is the same as translating units to the right and unit up. Translating along is the same as translating units to the left and unit down. Perform both compositions and compare the resulting images.
b Consider the vectors that connect the vertices and their images. Are they all parallel and with the same magnitude? Do all of them have the same direction?

Solution

a Translating along is equivalent to translating it units to the right and unit up. Similarly, translating along is equivalent to translating it units to the left and unit down.
Translating Along Is Equivalent To
Translating units to the right and unit up.
Translating units to the left and unit down.
To determine the correct order, both compositions should be tried. First, perform the translation along followed by the translation along
Translation along u first followed by the translation along v
As it can be seen, the above composition maps onto Next, perform the translation along followed by the translation along
Translation along v first followed by the translation along u
The last composition also mapped onto Consequently, the order in which the translations are applied is insignificant. This implies that the composition of translations is commutative.
b The fact that the corresponding sides of both polygons are parallel suggests that can be the image of under a single translation. To confirm this, draw the vectors connecting corresponding vertices.
Vectors connecting each vertex with its image

All the vectors drawn seem to be parallel and with the same magnitude. Even more, they all have the same direction. This could be checked by finding the component form of each vector.

Vertex Image Vector Component Form

As the table shows, all the vectors connecting a preimage with its image have the same component form. This confirms that the vectors are parallel and have the same magnitude and direction. Consequently, a translation along maps onto

Closure

Translations in the Real World

The two conclusions obtained in the previous example are not a coincidence. In fact, these are general results when performing a composition of translations.

The composition of translation is commutative and its translation vector is the sum of the corresponding translation vectors

Be aware that a composition of transformations might involve translations and rotations. This combination can produce interesting images.

In interior design, it is pretty common to see designs consisting of a single preimage and its images under different transformations such as translations and rotations. Below, two different kitchen tile designs are made using just four right triangles.
Showing two different kitchen tile designs by translating four right triangles to different positions of a square
Although four triangles were used, notice that three of them can be seen as the image of the fourth triangle after three different rotations. To finish this lesson, try creating your own design by performing translations on the given triangles.
Applet to create its own kitchen tile design
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