2. Transformations and Rigid Motions of Figures
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Chapter 1
2.
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.
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| | 13 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Transformations and Rigid Motions of Figures
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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.
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Transforming Polygons
In the applet below, four transformations can be applied to different polygons. Each transformation will affect the polygons in different ways.
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Mapping Polygons With Transformations
Consider the following pair of triangles. Here, △ ABC can be translated and rotated around point P. Is there a way of combining these two transformations so that △ ABC is mapped onto △ XYZ? If so, describe the steps used.
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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) → (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.
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Example
Translating a Polygon
Consider a transformation T that translates a polygon. Below, its effect on polygon P is shown.
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' |
|---|---|
| AB= 2.1 cm | A'B'= 2.1 cm |
| BC= 2.4 cm | B'C'= 2.4 cm |
| CD= 2 cm | C'D'= 2 cm |
| AD= 1.5 cm | A'D'= 1.5 cm |
| m∠ A=74^(∘) | m∠ A'=74^(∘) |
| m∠ B= 92^(∘) | m∠ B'= 92^(∘) |
| m∠ C=60^(∘) | m∠ C'=60^(∘) |
| m∠ D= 134^(∘) | m∠ D'= 134^(∘) |
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.
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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.
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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 △ JKL 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 △ JKL | Dimensions of △ J'K'L' |
|---|---|
| JK= 2.2 cm | J'K'= 2.2 cm |
| KL= 3.5 cm | K'L'= 3.5 cm |
| JL= 1.7 cm | J'L'= 1.7 cm |
| m∠ J= 127^(∘) | m∠ J'= 127^(∘) |
| m∠ K= 23^(∘) | m∠ K'= 23^(∘) |
| m∠ L= 30^(∘) | m∠ L'= 30^(∘) |
As the table shows, both △ JKL 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.
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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. AB=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.
Extra
A Composition of Rigid MotionsA composition of a translation, a rotation, and a reflection is also a rigid motion because it preserves the distance between any two points on the preimage.
Notice that this type of transformation also preserves the angle measures of the figure. However, its position and orientation can sometimes be affected.
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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.
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Answer
Ali is correct, T is a rigid motion. Magdalena could be confused because the transformation changes 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, AB and A'B' will be in the same row.
| Dimensions of ABCD | Dimensions of A'B'C'D' |
|---|---|
| AB= 2.8 cm | A'B'= 2.8 cm |
| BC= 2.2 cm | B'C'= 2.2 cm |
| CD=1.8 cm | C'D'=1.8 cm |
| AD= 1.9 cm | A'D'= 1.9 cm |
| m∠ A= 72^(∘) | m∠ A'= 72^(∘) |
| m∠ B=80^(∘) | m∠ B'=80^(∘) |
| m∠ C= 88^(∘) | m∠ C'= 88^(∘) |
| m∠ D= 121^(∘) | m∠ D'= 121^(∘) |
As the table shows, both quadrilaterals ABCD 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.
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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.
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.
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Example
Applying Different Transformations
Consider a triangle with vertices A(-2,-1), B(2,1), and C(0,2).
a Draw the image of △ ABC after increasing the y-coordinate of each point by 3 units.
b Draw the image of △ ABC after doubling the x-coordinate of each point.
c Draw the image of △ ABC after changing the sign of the y-coordinate of each point.
d Draw the image of △ ABC after taking the absolute value of the y-coordinate of each point.
e Which of the above options represent rigid motions?
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Answer
a 
b 
c 
d 
e A and C are rigid motions. The operation given in part D is not even a transformation.
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 △ ABC.
| 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 △ ABC. This can be seen in the diagram below.
Consequently, the transformation performed on △ ABC represents a translation 3 units up. It can be checked that both △ ABC 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 △ ABC is a horizontal stretch. It can be seen that the sides of △ A'B'C' are longer than the sides of △ ABC. 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 △ ABC. This can be seen in the diagram below.
In conclusion, the transformation performed on △ ABC is a reflection in the x-axis. It can be checked that both △ ABC 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 △ ABC 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 △ ABC 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.
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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.
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Example
Applying a Composition of Transformations
Magdalena and Ali were each given a rigid motion that they have to apply to pentagon ABCDE 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 90^(∘) counterclockwise rotation about the origin.
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?
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What are the coordinates of C'' when the transformations are not applied in the correct order?
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Hint
If the point (x,y) is rotated 90^(∘) 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).
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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.
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Transformations and Rigid Motions of Figures
Exercise 2.1
Which of the polygons, B through E, are transformations of A using only rigid motions?
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To solve this exercise, we should first understand what transformations that are rigid motion transformations.
Rigid Motion Transformations
For polygons that undergo rigid motions, shape and size are preserved. This means the image and preimage are congruent. This is why rigid motions also are known as congruence transformations. There are three types of rigid motions.
Not Rigid Motion Transformations
A dilation changes the size of a figure but not its shape. Therefore, this type of transformation is known as a similarity transformation.
Which Polygons are Dilated?
In order to pick out the polygons that have only undergone rigid motions, we need to exclude polygons that have been dilated. Notice that of the four sides of polygon A, we can make out the length of one of its sides, namely the horizontal one.
Let's measure the length of the corresponding side in each of the other polygons. Any polygon where this side is 2 units long has only undergone rigid motions.
Conclusion
As we can see from the diagram, D and E have sides that are greater than corresponding sides in polygon A. This indicates that besides undergoing transformations, they have also undergone a dilation. Therefore, B and C are the only figures which have undergone rigid motions only.
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To find the coordinates of B, we have to undo the transformations that carried △ ABC onto △ A''B''C''. Note that we have to undo the transformations in the correct order. Since △ ABC first underwent a translation and then a rotation, we must first perform a rotation.
Rotation
Since we rotated the triangle by 90^(∘) clockwise about the origin, we undo this transformation by rotating 90^(∘) counterclockwise about the origin. The following applies for such a transformation. Rotation of90^(∘) counterclockwise about the origin (x,y) → (- y, x) With this information, we can figure out the coordinates of △ A'B'C'.
| Point | (x,y) | (- y,x) |
|---|---|---|
| A | (1,4) | (-4,1) |
| B | (3,1) | (-1,3) |
| C | (7,4) | (-4,7) |
Now we can draw △ A'B'C' in the coordinate plane.
Translation
Next, we want to undo the translation which brought △ ABC to △ A'B'C'. Since we did a horizontal translation of 3 units to the right, and a vertical translation of 1 unit up, we undo this by performing the following translation on △ A'B'C'. (x,y) → (x-3,y-1) Let's perform this translation on △ A'B'C'.
The coordinates of B are (-4,2).
This time, we must start with the translation. As we explained in Part A, we must move the vertices of △ ABC by 3 units to the left and 1 unit down to undo the translation.
Notice that B' ended up at the origin. Therefore, when we rotate △ A'B'C' the point B' will not move. This means B will have the coordinates (0,0).
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Solution
Consider the following figure.
Pair the correct transformation which the top pieces have to undergo to create two full rows at the bottom. i. & (x,y) → (x+3,y-6) [1em] ii. & (x,y) → (x,y-6), & rotation of 180^(∘) [0.8em] iii & (x,y) → (x+3,y-4), & rotation of 90^(∘) For the rotations, you are free to pick the center of rotation. Also, the transformations do not necessarily have to come in the given order.
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The only way to create two full rows using the given pieces is by placing them like below.
Piece B
If we compare the original piece to the transformed piece, we see that this piece has been translated down and to the right. Since the orientation is the same, these translations are the only transformations that B underwent. Let's count the number of steps the piece must travel to end up at the right spot.
As we can see, we have to travel a total of 6 units down and 3 units to the right. This corresponds to i. i. & (x,y) → (x+3,y-6)
Piece C
In order to make C fit in its place, we will rotate it 180^(∘), and then translate the piece until it ends up between the blue and yellow piece. The only combination of transformations that contain a 180^(∘) rotation is ii. ii. & (x,y) → (x,y-4), & and a rotation of 180^(∘) Notice that we translate C to the left to give it the correct horizontal position, but the given translation does not include a shift to the left. By choosing an appropriate center of rotation, we can rotate C into the correct horizontal position.
As we can see, the transformation described by ii places piece C at the correct position.
Piece A
To make the polygon fit using transformation iii, we must choose an appropriate center of rotation.
We now have two full rows.
Conclusion
Now we can pair the transformations with the correct piece. &A → iii &B → i &C → ii
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Transformations and Rigid Motions of Figures
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