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Congruence, Proof, and Constructions

Transformations and Rigid Motions of Figures

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.

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, can be translated and rotated around point Is there a way of combining these two transformations so that is mapped onto 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 For that function, there is no real value of for which the output is 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, and orientation of a figure. The original figure is called the preimage and the produced figure is called the image of the transformation. To denote the image of a labeled point, a prime symbol is added next to the previous label.

A triangle and its image under a transformation

Transformations are sometimes expressed as a mapping. That is, it maps the inputs to the outputs. Note that the input can be a single point.

Here, is the transformation, are the coordinates of the points of the preimage, and are the coordinates of the points of the image.

Example

Translating a Polygon

Consider a transformation that translates a polygon. Below, its effect on polygon 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

Answer

Both and have the same side lengths and angle measures. For this reason, it can be established that does not affect the shape of which means that 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 and

Solution

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

Measuring the sides of both polygons

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

Measuring the angles of both polygons

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

Dimensions of Dimensions of
cm cm
cm cm
cm cm
cm cm

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

The transformation 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 on the polygon. By transforming different polygons, the same conclusion can be obtained.

Example

Rotating a Polygon

A transformation rotates a polygon around a fixed point Magdalena and Ali are trying to determine whether or not modifies the shape of the polygon. Magdalena thinks that does not change the shape, while Ali believes it does. They decided to apply 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 and

Solution

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

Measuring the side lengths of both triangles

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

Measuring the side lengths of both triangles

The dimensions found can be summarized in a table.

Dimensions of Dimensions of
cm cm
cm cm
cm cm

As the table shows, both and have the same side lengths and angle measures. Therefore, does not affect the shape of the polygon. Consequently, Magdalena is correct.

The transformation 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 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 of the points of the preimage.

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.

Rigid motions applied to a polygon

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

Answer

Ali is correct, 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 modifies the orientation of the quadrilateral.

Solution

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

Measuring the side lengths of both quadrilaterals

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

Measuring the angles of both quadrilaterals

In the table, all the dimensions found are summarized putting on the same row the dimensions of corresponding parts. For example, and will be in the same row.

Dimensions of Dimensions of
cm cm
cm cm
cm cm
cm cm

As the table shows, both quadrilaterals and have the same side lengths and angle measures. This means that does not affect the shape of the polygon. Consequently, Ali is correct.

The transformation is a rigid motion.

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

Comparing the Orientation of the preimage and the image

This fact could be what made Magdalena think that 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.
A triangle and its reflected image

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 and

Triangle ABC in the coordinate plane
a Draw the image of after increasing the coordinate of each point by units.
b Draw the image of after doubling the coordinate of each point.
c Draw the image of after changing the sign of the coordinate of each point.
d Draw the image of after taking the absolute value of the coordinate of each point.
e Which of the above options represent rigid motions?

Answer

a
Image of Triangle ABC
b
Image of Triangle ABC
c
Image of Triangle ABC
d
Image of Triangle ABC
e A and C are rigid motions. The operation given in part D is not even a transformation.

Hint

a Add to each coordinate.
b Multiply each coordinate by
c Multiply each coordinate by
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 units to the coordinate of each point while keeping the same coordinate. Start by performing this operation on the vertices of
Original Vertex Add to Each Coordinate New Vertex
As can be seen, each vertex is translated units up. The same will happen with all the points of This can be seen in the diagram below.
Triangle ABC and its image
Consequently, the transformation performed on represents a translation units up. It can be checked that both and have the same dimensions. Therefore, this operation represents a rigid motion.
b Doubling each coordinate is the same as multiplying them by This time, the coordinate remains unchanged.
Original Vertex Double each Coordinate New Vertex
Notice that and moved away from the axis while stayed in the same place. In the following diagram, it can be seen the effect of this transformation.
Triangle ABC and its image
The transformation performed on is a horizontal stretch. It can be seen that the sides of are longer than the sides of Also, the angle measures have changed. Consequently, this transformation is not a rigid motion.
c Changing the sign of each coordinate is the same as multiplying them by Here, the coordinates remain the same.
Original Vertex Multiply Each Coordinate by New Vertex
Notice that each vertex was reflected across the axis. The same happens for all the points of This can be seen in the diagram below.
Triangle ABC and its image
In conclusion, the transformation performed on is a reflection in the axis. It can be checked that both and 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 Coordinate New Vertex
Notice that the vertices and remained at the same place, while was reflected across the axis. This means that every point of whose coordinate is negative is reflected in the axis. Conversely, if the coordinate is positive, the point stays at the same place.
Triangle ABC and its image
As the graph above illustrates, the shape of was changed. Therefore, this operation is not a rigid motion. The operation folded the part of the triangle that is below the axis along the axis. Consider also the points and
Highlighting points P and Q

These two points have the same image under this operation — and 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.

Diagram for the composition of transformations T_2(T_1(F))= T_2(F_1)=F_2
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 one after the other in a particular order.

  • Magdalena has to perform a translation of units to the right and down.
  • Ali has to perform a counterclockwise rotation about the origin.
Initial Pentagon

Their teacher said that after applying both transformations in the correct order, the image of is Who should apply the transformation first?

What are the coordinates of when the transformations are not applied in the correct order?

Hint

If the point is rotated counterclockwise about the origin, its new coordinates will be

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.
Interactive Graph to perform the transformations
The table contains and — the images of and — after each composition is performed.
Preimage Translation Followed by Rotation Rotation Followed by Translation

The teacher said that performing the transformations in the correct order maps onto 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 is

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.

Mount-Hood-reflected-in-Mirror-Lake-2.jpg

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.
Prototype-Pontiac-race-car-2.jpg
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