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2. Transformations and Rigid Motions of Figures

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Chapter 1

Transformations and Rigid Motions of Figures

Transformations in geometry involve changing the position, size, or orientation of shapes. Rigid motions, a subset of transformations, only change the position and orientation without altering the size or shape. Examples of rigid motions include translations, rotations, and reflections. In the context of the coordinate plane, transformations can be visualized as mappings from one set of points to another. For instance, reflecting a shape across a vertical line will only change the x-coordinates of its points. The study of these transformations is crucial for understanding various geometric concepts and their applications in the real world, such as in car movements or reflections in mirrors.

Lesson Settings & Tools

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

Image Credits

Slide of 13

In Algebra, a function is a relation in which each input value is mapped to exactly one output value. This notion of function can be extended to Geometry. However, the inputs and outputs are not going to be numbers but points.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Congruent Segments
Congruent Angles
Functions
Coordinate plane

Explore

Transforming Polygons

In the applet below, four transformations can be applied to different polygons. Each transformation will affect the polygons in different ways.

Different Types of Transformations applied to geometric polygons

Compare the original polygon and the polygon obtained after applying a transformation. In a few words, describe the effect of each transformation.

Explore

Mapping Polygons With Transformations

Consider the following pair of triangles. Here,

△ A B C

can be translated and rotated around point

P .

Is there a way of combining these two transformations so that

△ A B C

is mapped onto

△ X Y Z ?

If so, describe the steps used.

A pair of triangles on a plane. One triangle needs to be move/rotated to match the other

Do not worry if the triangles cannot be mapped onto each other. Rather, consider the function

f (x) = x^{2} .

For that function, there is no real value of

x

for which the output is

- 1!

Remember this concept for later.

Discussion

Transformations as Functions

As stated at the beginning of this lesson, in Geometry, there are functions whose inputs and outputs are points. Such functions are called transformations.

Concept

Transformation

A transformation is a function that changes a figure in a particular way — it can change the position, size, or orientation of a figure. The original figure is called the preimage and the figure produced is called the image of the transformation. A prime symbol is often added to the label of a transformed point to denote that it is an image.

Transformations are sometimes expressed as a mapping because they map the inputs to the outputs. Note that an input can be a single point.

T (x, y) \to (x^{'}, y^{'})

Here,

T

is the transformation,

x

and

y

are the coordinates of the point of the preimage, and

x^{'}

and

y^{'}

are the coordinates of the point of the image.

Example

Translating a Polygon

Consider a transformation

T

that translates a polygon. Below, its effect on polygon

P

is shown.

A polygon and its image under the transformation T_1

Compare the side lengths and angle measures between the preimage and the image. Based on this observation, what can be said about

T ?

Answer

Both $P$ and $P^{'}$ have the same side lengths and angle measures. For this reason, it can be established that $T$ does not affect the shape of $P,$ which means that $T$ preserves the side lengths and angle measures.

Hint

Use a ruler to find the side lengths of both polygons. To find the angle measures, use a protractor. Make a table comparing the dimensions of $P$ and $P^{'} .$

Solution

With the aid of a ruler, the side lengths of both polygons can be found.

Additionally, with the help of a protractor, the angle measures of both polygons can be determined.

It is beneficial to summarize the information about the side lengths and angle measures of both polygons in a table.

Dimensions of $P$	Dimensions of $P^{'}$
$A B = 2.1$ cm	$A^{'} B^{'} = 2.1$ cm
$B C = 2.4$ cm	$B^{'} C^{'} = 2.4$ cm
$C D = 2$ cm	$C^{'} D^{'} = 2$ cm
$A D = 1.5$ cm	$A^{'} D^{'} = 1.5$ cm
$m ∠ A = 7 4^{\circ}$	$m ∠ A^{'} = 7 4^{\circ}$
$m ∠ B = 9 2^{\circ}$	$m ∠ B^{'} = 9 2^{\circ}$
$m ∠ C = 6 0^{\circ}$	$m ∠ C^{'} = 6 0^{\circ}$
$m ∠ D = 13 4^{\circ}$	$m ∠ D^{'} = 13 4^{\circ}$

As it can be seen in the table, both polygons $P$ and $P^{'}$ have the same side lengths and angle measures. Therefore, it can be concluded that $T$ does not affect the shape of $P .$ In fact, $T$ only affects the position of the polygon.

The transformation $T$ preserves side lengths and angle measures.

It is important to note that the conclusion does not depend on the polygon but on the effect of $T$ on the polygon. By transforming different polygons, the same conclusion can be obtained.

Example

Rotating a Polygon

A transformation

T

rotates a polygon around a fixed point

Q .

Magdalena and Ali are trying to determine whether or not

T

modifies the shape of the polygon. Magdalena thinks that

T

does not change the shape, while Ali believes it does. They decided to apply

T

to a triangle.

A Triangle and its image under the transformation T

Compare the side lengths and angle measures between the preimage and the image. Who is correct?

Hint

For each triangle, use a ruler and a protractor to find the side lengths and angle measures, respectively. Make a table comparing the dimensions of $△ J K L$ and $△ J^{'} K^{'} L^{'} .$

Solution

With the aid of a ruler, the side lengths of both triangles can be found.

Next, with the help of a protractor, the angle measures of both triangles can be found.

The dimensions found can be summarized in a table.

Dimensions of $△ J K L$	Dimensions of $△ J^{'} K^{'} L^{'}$
$J K = 2.2$ cm	$J^{'} K^{'} = 2.2$ cm
$K L = 3.5$ cm	$K^{'} L^{'} = 3.5$ cm
$J L = 1.7$ cm	$J^{'} L^{'} = 1.7$ cm
$m ∠ J = 12 7^{\circ}$	$m ∠ J^{'} = 12 7^{\circ}$
$m ∠ K = 2 3^{\circ}$	$m ∠ K^{'} = 2 3^{\circ}$
$m ∠ L = 3 0^{\circ}$	$m ∠ L^{'} = 3 0^{\circ}$

As the table shows, both $△ J K L$ and $△ J^{'} K^{'} L^{'}$ have the same side lengths and angle measures. Therefore, $T$ does not affect the shape of the polygon. Consequently, Magdalena is correct.

The transformation $T$ preserves side lengths and angle measures.

It is important to note that the conclusion does not depend on the polygon chosen. Instead, the conclusion depends on the effect of $T$ on the polygon. By using different polygons, the same conclusion can be obtained.

Discussion

Rigid Motions and Their Properties

Notice that the two transformations previously studied share the same property. The transformations neither affected the size nor the shape of the polygon. Still, they did affect the polygon's position on the plane. These types of transformations are called rigid motions.

Concept

Rigid Motion

A rigid motion, or isometry, is a transformation that preserves the distance between any two points on the preimage.

A B = A^{'} B^{'}

The following diagram displays two logos. The logo with the points

A

and

B

is the preimage and the logo with the points

A^{'}

and

B^{'}

is the image. The image is the result of a rigid motion because the distances between all points are preserved.

Two logos of Mathleaks with the letters ML and points A, B and their images A' and B'

Because rigid motions preserve distances, there are two properties that can be inferred from the definition.

Rule

Properties of Rigid Motions

A rigid motion preserves the side lengths and angle measures of a polygon. As a result, a rigid motion maintains the exact size and shape of a figure. Still, a rigid motion can affect the position and orientation of the figure.

Proof

A rigid motion preserves the side lengths of a polygon because, by definition, the distance between the vertices do not change.
It is accepted without a proof that rigid motions also preserve angle measures.

Example

Reflecting a Polygon

Ali and Magdalena are now working with a new transformation

T .

This transformation reflects a figure across a line as if the line was a mirror. In one of the attempts, they used a quadrilateral.

A quadrilateral and its image under the transformation T

After seeing the result, Ali concluded that

T

is a rigid motion. However, Magdalena said it is not. Who is correct? Why does Magdalena think that

T

is not a rigid motion?

Answer

Ali is correct, $T$ is a rigid motion. Magdalena could be confused because the transformation changes the orientation of the quadrilateral.

Hint

Verify whether or not the preimage and the image have the same side lengths and angle measures. Notice that $T$ modifies the orientation of the quadrilateral.

Solution

Using a ruler, the side lengths of both quadrilaterals can be found.

Next, with a protractor, the angle measures can be found.

In the table, all the dimensions found are summarized putting on the same row the dimensions of corresponding parts. For example, $A B$ and $A^{'} B^{'}$ will be in the same row.

Dimensions of $A B C D$	Dimensions of $A^{'} B^{'} C^{'} D^{'}$
$A B = 2.8$ cm	$A^{'} B^{'} = 2.8$ cm
$B C = 2.2$ cm	$B^{'} C^{'} = 2.2$ cm
$C D = 1.8$ cm	$C^{'} D^{'} = 1.8$ cm
$A D = 1.9$ cm	$A^{'} D^{'} = 1.9$ cm
$m ∠ A = 7 2^{\circ}$	$m ∠ A^{'} = 7 2^{\circ}$
$m ∠ B = 8 0^{\circ}$	$m ∠ B^{'} = 8 0^{\circ}$
$m ∠ C = 8 8^{\circ}$	$m ∠ C^{'} = 8 8^{\circ}$
$m ∠ D = 12 1^{\circ}$	$m ∠ D^{'} = 12 1^{\circ}$

As the table shows, both quadrilaterals $A B C D$ and $A^{'} B^{'} C^{'} D^{'}$ have the same side lengths and angle measures. This means that $T$ does not affect the shape of the polygon. Consequently, Ali is correct.

The transformation $T$ is a rigid motion.

Although $T$ is a rigid motion, notice that the orientation of the preimage and the image are not the same. In the preimage, the vertices from $A$ to $D$ are positioned counterclockwise, while in the image, they are positioned clockwise.

This fact could be what made Magdalena think that $T$ is not a rigid motion. Notice that the transformations studied before preserve orientations.

Explore

Rigid Transformations

Consider a triangle and its image under a rigid motion drawn on a sketch pad. If the transformation is a translation or a rotation, by using a tracing paper the preimage can be mapped onto the image without peeling off the tracing paper from the sketch pad.

Using Tracing Paper to Translate/Rotate a Triangle onto its Image under a Translation/Rotation

Conversely, suppose that transformation is a reflection. In that case, it is not possible to map the preimage onto the image without peeling off the tracing paper from the sketch pad. The reason is that reflections change the orientation of the figures.

Mapping the preimage onto the image would necessitate folding the tracing paper along the line of reflection. However, by doing this the tracing paper peels off the sketch pad making a three-dimensional movement.

Folding a Tracing Paper to map a Triangle onto its Image under a reflection

For this reason, reflections are sometimes called rigid transformations instead of rigid motions. Although all the transformations studied so far were applied to polygons, keep in mind that transformations can be applied to any point on the coordinate plane. The reason for using geometric shapes is that they make it easier to see the effect produced by a transformation and whether the transformation is a rigid motion or not.

Example

Applying Different Transformations

Consider a triangle with vertices $A (- 2, - 1),$ $B (2, 1),$ and $C (0, 2) .$

The triangle ABC is defined by vertices labeled counterclockwise: A (-2, -1), B (2, 1), and C (0, 2).

a Draw the image of

△ A B C

after increasing the

y -

coordinate of each point by

3

units.

b Draw the image of

△ A B C

after doubling the

x -

coordinate of each point.

c Draw the image of

△ A B C

after changing the sign of the

y -

coordinate of each point.

d Draw the image of

△ A B C

after taking the absolute value of the

y -

coordinate of each point.

e Which of the above options represent rigid motions?

Answer

e A and C are rigid motions. The operation given in part D is not even a transformation.

Hint

a Add

3

to each

y -

coordinate.

b Multiply each

x -

coordinate by

2 .

c Multiply each

y -

coordinate by

- 1 .

d The absolute value of a negative number is its opposite.

e A rigid motion is a transformation that preserves side lengths and angle measures. Compare the shapes of the preimage and the image.

Solution

a This first operation requires adding

3

units to the

y -

coordinate of each point while keeping the same

x -

coordinate. Start by performing this operation on the vertices of

△ A B C .

Original Vertex	Add $3$ to Each $y -$ Coordinate	New Vertex
$A (- 2, - 1)$	$(- 2, - 1 + 3)$	$A^{'} (- 2, 2)$
$B (2, 1)$	$(2, 1 + 3)$	$B^{'} (2, 4)$
$C (0, 2)$	$(0, 2 + 3)$	$C^{'} (0, 5)$

As can be seen, each vertex is translated

3

units up. The same will happen with all the points of

△ A B C .

This can be seen in the diagram below.

Consequently, the transformation performed on

△ A B C

represents a translation

3

units up. It can be checked that both

△ A B C

and

△ A^{'} B^{'} C^{'}

have the same dimensions. Therefore, this operation represents a rigid motion.

b Doubling each

x -

coordinate is the same as multiplying them by

2 .

This time, the

y -

coordinate remains unchanged.

Original Vertex	Double each $x -$ Coordinate	New Vertex
$A (- 2, - 1)$	$(2 (- 2), - 1)$	$A^{'} (- 4, - 1)$
$B (2, 1)$	$(2 (2), 1)$	$B^{'} (4, 1)$
$C (0, 2)$	$(2 (0), 2)$	$C^{'} (0, 2)$

Notice that

A

and

B

moved away from the

y -

axis while

C

stayed in the same place. In the following diagram, it can be seen the effect of this transformation.

The transformation performed on

△ A B C

is a horizontal stretch. It can be seen that the sides of

△ A^{'} B^{'} C^{'}

are longer than the sides of

△ A B C .

Also, the angle measures have changed. Consequently, this transformation is not a rigid motion.

c Changing the sign of each

y -

coordinate is the same as multiplying them by

- 1 .

Here, the

x -

coordinates remain the same.

Original Vertex	Multiply Each $y -$ Coordinate by $- 1$	New Vertex
$A (- 2, - 1)$	$(- 2, - 1 (- 1))$	$A^{'} (- 2, 1)$
$B (2, 1)$	$(2, 1 (- 1))$	$B^{'} (2, - 1)$
$C (0, 2)$	$(0, 2 (- 1))$	$C^{'} (0, - 2)$

Notice that each vertex was reflected across the

x -

axis. The same happens for all the points of

△ A B C .

This can be seen in the diagram below.

In conclusion, the transformation performed on

△ A B C

is a reflection in the

x -

axis. It can be checked that both

△ A B C

and

△ A^{'} B^{'} C^{'}

have the same dimensions. Therefore, this transformation is a rigid motion.

d Recall that the absolute value of a positive number is the same number. Conversely, the absolute value of a negative number is its opposite.

Original Vertex	Taking the Absolute Value of Each $y -$ Coordinate	New Vertex
$A (- 2, - 1)$	$(- 2, ∣ - 1 ∣)$	$A^{'} (- 2, 1)$
$B (2, 1)$	$(2, ∣ 1 ∣)$	$B^{'} (2, 1)$
$C (0, 2)$	$(0, ∣ 2 ∣)$	$C^{'} (0, 2)$

Notice that the vertices

B

and

C

remained at the same place, while

A

was reflected across the

x -

axis. This means that every point of

△ A B C

whose

y -

coordinate is negative is reflected in the

x -

axis. Conversely, if the

y -

coordinate is positive, the point stays at the same place.

As the graph above illustrates, the shape of

△ A B C

was changed. Therefore, this operation is not a rigid motion. The operation folded the part of the triangle that is below the

x -

axis along the

x -

axis. Consider also the points

P (- 1, - 0.5)

and

Q (- 1, 0.5) .

These two points have the same image under this operation — $P^{'} (- 0.5, 0.5)$ and $Q^{'} (- 0.5, 0.5) .$ This means that this is not a one-to-one operation. Consequently, it is not even a transformation!

e As previously stated, the transformations in parts A and C are rigid motions.

Discussion

Combining Transformations

Consider the fact that two or more functions can be applied one after the other to an input. Similarly, two or more transformations can be applied one after the other to a preimage.

Concept

Composition of Transformations

A composition of transformations, or sequence of transformations, is a combination of two or more transformations. In a composition, the image produced by the first transformation is the preimage of the second transformation. The notation is similar to the notation used for functions in algebra.

If the transformations are rigid motions, the composition is also a rigid motion.

When performing the composition of transformations, the order in which they are applied matters. In some cases, switching the transformations can lead to erroneous results.

Example

Applying a Composition of Transformations

Magdalena and Ali were each given a rigid motion that they have to apply to pentagon $A B C D E$ one after the other in a particular order.

Magdalena has to perform a translation of $2$ units to the right and $1$ down.
Ali has to perform a $9 0^{\circ}$ counterclockwise rotation about the origin.

An irregular pentagon with vertices A(-2,2), B(-3,-2), C(-1,-4), D(2,2), and E(2,-2) on a coordinate plane.

Their teacher said that after applying both transformations in the correct order, the image of $B (- 3, - 2)$ is $B^{''} (4, - 4) .$ Who should apply the transformation first?

What are the coordinates of

C^{''}

when the transformations are not applied in the correct order?

Hint

If the point $(x, y)$ is rotated $9 0^{\circ}$ counterclockwise about the origin, its new coordinates will be $(- y, x) .$

Solution

Both compositions should be tried to determine what is the correct order. First, perform the translation followed by the rotation. Then, perform the rotation followed by the translation.

The table contains

B^{''}

and

C^{''}

— the images of

B

and

C

— after each composition is performed.

Preimage	Translation Followed by Rotation	Rotation Followed by Translation
$B (- 3, - 2)$	$B^{''} (3, - 1)$	$B^{''} (4, - 4)$
$C (- 1, - 4)$	$C^{''} (5, 1)$	$C^{''} (6, - 2)$

The teacher said that performing the transformations in the correct order maps $B (- 3, - 2)$ onto $B^{''} (4, - 4) .$ This means that the correct order is the rotation followed by the translation. Therefore, Ali goes first. If the transformations are not performed in the correct order, the image of $C (- 1, - 4)$ is $C^{''} (5, 1) .$

Closure

Transformations in the Real World

Keep in mind that transformations transcend beyond paper and can be used for many purposes. In the real world, transformations occur everywhere. For example, in nature, lakes perform reflections of landscapes.

Also, different types of transformations can be seen by observing a car, either in rest or in movement.

A car moving in a straight line represents a translation.
The wheels of a moving car show rotations.
Rear-view and side-view mirrors make reflections.
Even car windows make reflections, and depending on the object's position, they can also make distortions.

Level 1

Level 2

Level 3

Transformations and Rigid Motions of Figures

Consider $△ A B C .$

Suppose $f$ is the mapping of the plane which moves each point $(x, y)$ to point $(x - 5, y),$ and $g$ is the mapping of the plane that moves each point $(x, y)$ to $(5 x, y) .$

Does

f

preserve distances and angles? Explain.

Does

g

preserve distances and angles? Explain.

Moving the point (x,y) to the point (x-5,y) is a translation of 5 units to the left. To obtain the corresponding points of A'B'C' we have to subtract 5 from the x-coordinate of each point.

Point	(x,y)	(x-5,y)
A	(-2,-3)	(-7,-3)
B	(-1,2)	(-6,2)
C	(2,-6)	(-3,-6)

Let's draw the translated triangle along with the preimage.

As we can see, the translated triangle has the same shape as its preimage. The sides also have the same length. Therefore, the transformation has preserved both distances and angles.

In Part B, we take the point (x,y) to the point (5x,y). This means the x-coordinate of each point gets multiplied by 5.

Point	(x,y)	(5x,y)
A	(-1,2)	(-5,2)
B	(-1,2)	(-5,2)
C	(2,-6)	(10,-6)

Let's sketch the transformed triangle and its preimage.

As we can see, neither shape nor distances are preserved. We know this because in △ ABC the longest side is BC. But in △ A''B''C'', the side A''C'' is the longest.

Which transformation(s) was used on each pair of shapes below. If more than one transformation was used, include the least number of transformations needed to map the shapes onto each other.

Two quadrilaterals that are reflections of each other.

Two quadrilaterals separated by a translation and a rotation.

Examining the diagram, we see that the shapes are mirror images of each other. This means we can map one shape onto the other by reflecting it in a line halfway between two corresponding points.

Therefore, only one transformation, a reflection, is needed to map them onto each other.

As in Part A, the two shapes have different position. They also have a different orientation. To map one shape onto the other, two transformations need to be performed.

Therefore, we can make a translation and a rotation to map the figures onto each other.

Consider $△ A B C$ in the coordinate plane.

Triangle ABC drawn in a coordinate plane. The triangle has vertices in the points (-4,2), (0,3), and (-3,6)

Rotate

△ A B C

9 0^{\circ}

counterclockwise about the origin creating

A^{'} B^{'} C^{'} .

What are the coordinates of

C^{'} ?

Reflect

△ A B C

across the vertical line

x = 2

to create

△ A^{''} B^{''} C^{''} .

What are the coordinates of

A^{''} .

Translate

△ A B C

so that

A^{'''}

is at

(4, - 5) .

What are the coordinates of

B^{'''} ?

If the point (x,y) is rotated 90^(∘) counterclockwise about the origin, its new coordinates will be (- y,x). This means the x-coordinate of the preimage becomes the y-coordinate of the image and the opposite number of the y-coordinate of the preimage becomes the x-coordinate of the image. Let's formalize it as a transformation. Rotation of90^(∘) Counterclockwise About the Origin (x,y) → (- y, x) With this information, we can find the coordinates of the rotated figure.

Point	(x,y)	(- y,x)
A	(-4,2)	(-2,-4)
B	(0,3)	(-3,0)
C	(-3,6)	(-6,-3)

Now we can draw △ A'B'C' in the coordinate plane.

The coordinates of C' are (-6,-3).

Let's draw the vertical line x=2 in the coordinate plane.

To reflect the triangle across x=2, we will draw a perpendicular segment from each vertex to x=2. We can then extend these segments on the opposite side of x=2. When the extensions are congruent with each corresponding segment we have found the positions of the reflected points.

The coordinates of A'' are (8,2).

In order to translate A to (4,-5), we will move the point horizontally 8 steps to the right and vertically 7 steps down.

To complete the transformation, we must translate the remaining points in the same way.

The coordinates of B''' are (8,-4).

Consider the following four figures which show different transformation(s).

Four different transformations on triangle A

Which figure is a reflection in

x = a ?

Which figure is a reflection of

y = x ?

Which figure is a reflection of

y = b ?

The line x=a is a vertical line. Whenever points are reflected across a vertical line, the y-coordinates do not change. Only the x-coordinates are affected. Only diagram i displays this behavior.

When a point (a,b) is reflected in the line y=x, its image has the coordinates (b,a). The graph of y=x is a 45 ^(∘) upward sloping line. If we reflect our image across this line, we see that the reflection resembles diagram iii.

In Part A, we explained that when we reflect something in a vertical line only the x-coordinates change. Conversely, if we reflect something in a horizontal line, only the y-coordinates change. We see this behavior in diagram ii.

Consider the following translation.

(x, y) \to (x - 8, y + 4)

What is the image of

A (2, 6) ?

What is the image of

B (- 1, 5) ?

What is the preimage of

C^{'} (- 3, - 10) ?

What is the preimage of

D^{'} (4, - 3) ?

To determine the location of A', point A must undergo the given transformation.

Point	(x,y)	(x-8,y+4)
A	(2,6)	(-6,10)

As we can see, the image after the translation is the point A'(-6,10).

As in Part A, we must perform the given transformation on B to obtain the coordinates of B'.

Point	(x,y)	(x-8,y+4)
B	(-1,5)	(-9,9)

As we can see, the point's coordinates after the translation are B'(-9,9).

This time, we want to determine the preimage of C'. This means we have to undo the given transformation. Therefore, we must add 8 to the x-coordinate and subtract 4 from the y-coordinate.

Point	(x,y)	(x+8,y-4)
C'	(-3,-10)	(5,-14)

By undoing the translation we have found the coordinates of C. C(5,- 14)

As in Part C, we must undo the given transformation to obtain the coordinates of D.

Point	(x,y)	(x+8,y-4)
D'	(4,-3)	(12,-7)

Undoing the translation gave us the coordinates of D. D(12,-7)

Transformations and Rigid Motions of Figures

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