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5. Translations of Figures in a Plane

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Chapter 1

5.

Translations of Figures in a Plane

This lesson focuses on the concept of translation in geometry, which involves moving geometric figures in a plane without altering their shape, size, or orientation. This is particularly useful in fields like engineering, architecture, and design where precise movement of shapes is required. For instance, in architecture, you might need to move a window design from one wall to another while maintaining its dimensions and orientation. The lesson provides exercises and examples to help you understand how to perform these translations effectively.

Lesson Settings & Tools

	12 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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Slide of 12

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

v

can be set. Then,

△ A B C

can be translated along the vector.

Once the translation is performed, is there any relationship between

\overline{A A^{'}},

\overline{B B^{'}},

\overline{C C^{'}},

and

v ?

Discussion

Properties of Translations

From the exploration, the following conclusions can be drawn.

The lengths of $\overline{A A^{'}},$ $\overline{B B^{'}},$ and $\overline{C C^{'}}$ are equal to the magnitude of $v .$
The three segments $\overline{A A^{'}},$ $\overline{B B^{'}},$ and $\overline{C C^{'}}$ are parallel to $v .$

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

\overline{A A^{'}},

\overline{B B^{'}},

and

\overline{C C^{'}}

are equal to the magnitude of

v .

Also,

\overline{A A^{'}},

\overline{B B^{'}},

and

\overline{C C^{'}}

are parallel to

v .

Therefore, the same two conclusions written above apply to this diagram.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ A A^{'} = ∣ v ∣ B B^{'} = ∣ v ∣ C C^{'} = ∣ v ∣ and ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ \overline{A A^{'}} ∥ v \overline{B B^{'}} ∥ v \overline{C C^{'}} ∥ v

However,

△ A^{'} B^{'} C^{'}

is not a translation of

△ A B C .

This is because, although vectors

B B^{'}

and

C C^{'}

have the same direction as

v,

the vector

A A^{'}

does not have the same direction as

v .

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 $A A^{'},$ $B B^{'},$ and $C C^{'}$ have the same direction as $v .$

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 of Geometric Objects

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 $v$ maps every point $A$ in the plane onto its image $A^{'}$ such that the following statements hold true.

The length of $\overline{A A^{'}}$ and the magnitude of $v$ are equal.
Segment $\overline{A A^{'}}$ and $v$ are parallel.
Vectors $A A^{'}$ and $v$ have the same direction.

These three properties imply that the quadrilateral formed by

A,

A^{'},

the tip of

v,

and the tail of

v

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

v

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

∠ A B C

and the length of its sides can be set. Also, the magnitude and direction of

v

can be defined. Once done,

∠ A B C

can be translated along

v .

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 $\overline{A B}$ and $\overline{A^{'} B^{'}} ?$
Do $\overline{B C}$ and $\overline{B^{'} C^{'}}$ have the same relationship as $\overline{A B}$ and $\overline{A^{'} B^{'}} ?$

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 $P$ shown in the middle of the diagram below. The other polygons are images of $P$ after different transformations.

What polygons are the image of $P$ after a translation?

Answer

Only polygons $P_{4},$ $P_{6},$ and $P_{14}$ are the image of $P$ after a translation.

Hint

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

Solution

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

Therefore, the image of

P

after any translation will also look like an arrow pointing up. In the given diagram, it can be seen that only

P_{4},

P_{6},

and

P_{14}

satisfy this condition. Consequently, these polygons are images of

P

after a translation.

To obtain the other polygons,

P

must be translated and a rotated.

Example

Translations and Polygon's Vertices

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

Which of the squares

S_{1}

or

S_{2},

if either, is the image of

P Q R S

after a translation?

Hint

Pay close attention to the labels of the vertices. Remember that, for every preimage $A$ and image $A^{'},$ $\overline{A A^{'}}$ 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

S_{1}

and

S_{2}

are the image of

P Q R S

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

A

and its image

A^{'},

the following relations hold true.

The segment whose endpoints are $A$ and $A^{'},$ $\overline{A A^{'}},$ 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 $A$ and $A^{'},$ $\overline{A A^{'}},$ 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

P Q R S

and

S_{2},

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, $S_{2}$ is not the image of $P Q R S$ after a translation. Conversely, $S_{1}$ 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 $S_{1}$ is a translation of $P Q R S .$

Discussion

Translations Performed by Hand

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

To translate $△ A B C$ along $v$ follow the four steps below.

1

Draw the Line

ℓ

Containing

v

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

2

Draw Parallel Lines to

ℓ

Through Each Vertex

Draw parallel lines to $ℓ$ through each vertex.

3

Copy

v

Over the Lines Drawn

Over the three lines drawn in the previous step, using a compass copy vector

v

in such way that each vertex of

△ A B C

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 $A^{'},$ $B^{'},$ and $C^{'} .$

Extra

Alternative Method

By definition of translation, vectors $A A^{'}$ and $v$ are parallel and have the same magnitude and direction. This makes $P A A^{'} Q$ a parallelogram. With this in mind, the translation of $△ A B C$ can be performed as follows.

Copy the length of $v$ : open the compass to the length of $v .$ Then, with this setting, place the compass tip on each vertex of $△ A B C$ and draw an arc.

Copy the length from $P$ to each vertex: place the compass tip on $P$ and open the compass to $C .$ Then, with this setting, place the compass tip on $Q$ and make an arc that intersects the arc centered at $C .$ The point of intersection is $C^{'} .$
Here, $P C C^{'} Q$ 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 $A^{'},$ $B^{'},$ and $C^{'} .$

Discussion

Translations in the Coordinate Plane

In the coordinate plane, the component form of the translation vector

v

is closely related to the coordinates of the image of a point

P (a, b) .

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

P (a, b)

after a translation along the vector

v = ⟨ v_{1}, v_{2} ⟩

is the point

P^{'} (a + v_{1}, b + v_{2}) .

In other words, to find the image of

P (a, b)

, add

v_{1}

to the

x -

coordinate and

v_{2}

to the

y -

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

A A^{'} .

Pop Quiz

Practice Translations

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

Translate $△ A B C$ along the given vector $v .$
Write the component form of the translation vector that maps $△ A B C$ onto $△ A^{'} B^{'} C^{'} .$

To translate $△ A B C,$ place points $A^{'},$ $B^{'},$ and $C^{'}$ 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 $P_{1}$ and $P_{2} .$ Also, consider a pair of different translations. One translation along vector $u = ⟨ 5, 1 ⟩$ and the other along $v = ⟨ - 2, - 4 ⟩ .$

Pair of quadrilaterals in the coordinate plane

a In which order must the translations be applied to

P_{1}

so that it is mapped onto

P_{2} ?

Only one option is correct.

{"type":"choice","form":{"alts":["Along <\/span><\/span>u<\/span><\/span><\/span><\/span><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> first","Along <\/span><\/span>v<\/span><\/span><\/span><\/span><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> first","The order is insignificant"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}

b Is there a single translation that maps

P_{1}

onto

P_{2} ?

If so, write the component form of the translation vector.

Hint

a Translating along

u

is the same as translating

5

units to the right and

1

unit up. Translating along

v

is the same as translating

2

units to the left and

4

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

P_{1}

along

u = ⟨ 5, 1 ⟩

is equivalent to translating it

5

units to the right and

1

unit up. Similarly, translating

P_{1}

along

v = ⟨ - 2, - 4 ⟩

is equivalent to translating it

2

units to the left and

1

unit down.

Translating Along	Is Equivalent To
$u = ⟨ 5, 1 ⟩$	Translating $5$ units to the right and $1$ unit up.
$v = ⟨ - 2, - 4 ⟩$	Translating $2$ units to the left and $4$ unit down.

To determine the correct order, both compositions should be tried. First, perform the translation along

u

followed by the translation along

v .

As it can be seen, the above composition maps

P_{1}

onto

P_{2} .

Next, perform the translation along

v

followed by the translation along

u .

The last composition also mapped

P_{1}

onto

P_{2} .

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

P_{2}

can be the image of

P_{1}

under a single translation. To confirm this, draw the vectors connecting corresponding vertices.

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
$(- 4, 0)$	$(- 1, - 3)$	$⟨ - 1 - (- 4), - 3 - 0 ⟩$	$⟨ 3, - 3 ⟩$
$(- 1, 1)$	$(2, - 2)$	$⟨ 2 - (- 1), - 2 - 1 ⟩$	$⟨ 3, - 3 ⟩$
$(- 1, 2)$	$(2, - 1)$	$⟨ 2 - (- 1), - 1 - 2 ⟩$	$⟨ 3, - 3 ⟩$
$(- 3, 3)$	$(0, 0)$	$⟨ 0 - (- 3), 0 - 3 ⟩$	$⟨ 3, - 3 ⟩$

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 $⟨ 3, - 3 ⟩$ maps $P_{1}$ onto $P_{2} .$

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

Level 1

Level 2

Level 3

Translations of Figures in a Plane

Exercises

Translate point $P (2, 5)$ according to the vector. What are the coordinates of the resulting image?

⟨ 6, 0 ⟩

⟨ 5, - 1 ⟩

⟨ - 8, - 7 ⟩

The vector form tells us how we should move in the coordinate plane. Vector Form: ⟨ 6, 0 ⟩ Horizontal translation:& + 6units Vertical translation:& 0units Let's plot point P in a coordinate plane and move it according to the vector form. Notice that we only have to perform a horizontal translation to the right of 6 units.

The image of P has the coordinates P'(8,5).

Like in Part A, we will analyze the vector form to determine how to move P. Vector Form: ⟨ 5, -1 ⟩ Horizontal translation:& + 5units Vertical translation:& - 1unit Let's move P according to the vector form. This time we must perform both a horizontal and a vertical translation.

As we can see, the image of P has the coordinates P'(7,4).

Like in previous parts, we will determine how P must move according to the vector form. Vector Form: ⟨ -8, -7 ⟩ Horizontal translation:& -8units Vertical translation:& - 7units If we move the point 3 units to the left and 7 units down we find P'.

Examining the coordinate plane, we see that P' has the coordinates (-6,-2).

Translate the given point $P$ using vector $v .$ What are the coordinates of the image $P^{'}$ ?

A point P(3,5) and an upward pointing vector

A point P(6,8) and a downward pointing vector

Point P(10,4) and a vector for translating the point

To translate P in the direction of the vector, we will move v until its starting point has the same coordinates as P. At the end of v, we will then find P'.

The coordinates of P' are (8,6).

As in Part A, we will translate the vector so that it starts at P. The endpoint of the vector tells us where the coordinates of the image is.

The image P' has the coordinates (8,4).

As in previous parts, v must be translated until its starting point is at P. Then we can find the coordinates of P' at the end of v.

The coordinates of P' are (5,5).

In the following pattern, two figures represent translations of each other. Can you tell which ones?

a patterna of several different polygons of which two are translations of each other.

A translation is a rigid motion that preserves the lengths and angle measures of a figure. Therefore, what we are looking for are two shapes that have exactly the same shape and orientation. There are only two such shapes: E and G. These can be translated across the grid and map onto each other.

A construction worker wants to use her front-end loader to move some debris on a construction site. In relation to the entry point of the construction site $(0, 0),$ the debris is at $(11, 1) .$ The worker moves it through the construction site along the vectors $⟨ - 6, 3 ⟩,$ $⟨ - 3, 3 ⟩,$ and $⟨ 7, 2 ⟩ .$ This gets her to the spot where she unloads the debris.

What are the coordinates of the unloading point?

What single vector from the loading point to the unloading point is correct?

Let's start by plotting the point where the debris is.

Next, we will determine the vertical and horizontal movements for the three vectors, which we will label v_1, v_2, and v_3. v_1 = ⟨ - 6, 3 ⟩ Horizontal translation:& - 6units Vertical translation:& +3units v_2 = ⟨ - 3, 3 ⟩ Horizontal translation:& - 3units Vertical translation:& + 3units v_3 = ⟨ 7, 2 ⟩ Horizontal translation:& +7units Vertical translation:& +2units Now we can illustrate the movements of the front-end loader.

The debris is dropped off at (9,9).

From Part A, we know the points where the debris is picked up and where it is unloaded.

A single vector that describes the movement of the front-end loader is the vertical and horizontal difference between these points.

The vector can be written as ⟨ - 2,8 ⟩.

Which of the polygon(s), $A$ through $E,$ are translations of $Q ?$

A translation changes a polygon's position but preserves shape, size, and orientation. Examining the five polygons, we see that A has a different size and B has a different shape. Therefore, neither can be translations of Q. Let's remove them.

The remaining polygons all have the same shape and size. However, we see that E has a different orientation. It is facing the wrong direction. Therefore, E can not be a translation of Q either. Let's remove it.

Let's attempt to translate Q and make it map onto C and D. If it does, C and D are translations of Q.

As we can see, C and D are both translations of Q.

Translations of Figures in a Plane

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