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2. Rotations and Dilations

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Pre-Algebra /

Chapter 11

Rotations and Dilations

Student Learning Objectives:
Rotate geometric objects about a fixed point Dilate a geometric object

Why

This lesson covers geometric transformations like rotation and dilation. It explains how to rotate a point around another and how triangles can be rotated. Additionally, it introduces the concept of dilations, focusing on how to apply coordinate rules for dilation in geometry. Understanding these transformations is crucial in solving problems involving shapes, symmetry, and scaling in various mathematical contexts.

Lesson Settings & Tools

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

Image Credits

Rotations and Dilations

Slide of 16

This lesson explores two new transformations, rotations and dilations. Rotations move shapes around a central point, while dilations either enlarge or shrink geometric figures without changing their shape.

Catch-Up and Review

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

Explore

Rotating Points About a Fixed Point

In the following applet, points A, B, C, and D, are movable. Arrange them as desired, then use the slider bar to rotate the points around the central point P.

Points A, B, C and D that can be rotated around Point P, a slider that determines the rotation angle, and a length measurement tool

Use the measuring tool to fill in the following table with the distances between P and the image and preimage of each point.

PA		PA'
PB		PB'
PC		PC'
PD		PD'

What conclusions can be drawn from the table? Repeat the process with different arrangements of points as needed to see a pattern.

Explore

Rotating a Triangle About a Fixed Point

This time, use the slider bar to rotate triangle ABC about point P.

Then, use the measuring tool to determine the angle measures of ∠ APA', ∠ BPB', and ∠ CPC'.

m ∠ APA'
m ∠ BPB'
m ∠ CPC'

Change the triangle and perform the same procedure. What can be said about rotations? Does the conclusion depend on the position of P relative to the triangle?

Discussion

Rotation

A rotation is a transformation in which a figure is turned about a fixed point P. The number of degrees the figure rotates α ^(∘) is the angle of rotation. The fixed point P is called the center of rotation. Rotations map every point A in the plane to its image A' such that one of the following statements is satisfied.

If A is the center of rotation, then A and A' are the same point.
If A is not the center of rotation, then A and A' are equidistant from P, with ∠ APA' measuring α ^(∘).

Rotations are usually performed counterclockwise unless stated otherwise.

Notice that a 90^(∘) counterclockwise rotation is the same as a 270^(∘) clockwise rotation. Since rotations preserve side lengths and angle measures, they are rigid motions.

Discussion

Rotation of a Point Around Another Point

Rotations can be performed by hand with the help of a straightedge, a compass, and a protractor.

To rotate point A about point P by an angle of 130^(∘) measured counterclockwise, follow these five steps.

Draw PA

Using the straightedge, draw the segment connecting the center of rotation P and point A.

Place the Protractor

Place the center of the protractor on P and align it with PA.

The protractor is placed as illustrated above when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.

Mark the Desired Angle

Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at 130^(∘).

Draw a Ray

Using the straightedge, draw a ray with starting point P that passes through the mark made in the previous step.

Drawing a Ray starting a P that makes a 130 degree angle with segment PA

Draw Point A'

Place the compass tip on P and open it to the distance between P and A. Without changing this setting and keeping the point of the compass at P, draw a small arc centered at P that intersects the ray drawn before.

Measuring PA with the compass and making an arc that intersects the ray drawn before

The intersection of the ray and the arc is the image A' after the give rotation.

Notice that this method of construction has also confirmed that PA is congruent to PA'.

Example

Cyrptawheel

Vincenzo and Zosia are two friends who love exploring the world of puzzles. Within the pages of a book discussing transformations, they find a circular disk with letters on it. They decide to call the disk a Cyrptawheel. The Cryptawheel has a movable triangle with one vertex fixed at the center of the circular disk. To reveal its secrets, the friends need to rotate the triangle.

Circular paper with some letters on it and triangular paper with vertices pointing N and E

Two vertices of the triangle show the letters N and E. These are the first two letters of a clue. Rotate the triangle 180^(∘) clockwise around its fixed vertex to find the other two letters. Zosia wants to perform this rotation using a protractor to be mathematically sure. Help her in this task.

Hint

Start by rotating the vertices of the triangle. To do so, use a protractor to draw a 180^(∘) angle.

Solution

Zosia's goal is to find the image of the triangle after a 180^(∘) clockwise rotation about its fixed vertex. Labeling the vertices of the triangle will make this rotation process simpler.

The center of rotation is O and the angle of rotation is 180^(∘). Perform the rotation by rotating one point at a time. To rotate N, place the center of the protractor on O and align it with NO. Use the protractor to draw a ray that starts from O and makes a 180^(∘) angle with NO.

Then, mark a point N' on this ray so that ON' is the same length as ON. This is the image of N after the rotation. Since ON is the radius of the inner circle, N' should also be on that circle.

Repeat the same process for the vertex E to find E'. Since O is the center of rotation, O' will be in the same position as O.

Finally connect N', E', and O' to draw the image of NEO after the 180^(∘) rotation.

As shown, the image N' corresponds to P and E' corresponds to T. Therefore, the clue Vincenzo and Zosia is looking for is NEPT.

Extra

Visualizing the Rotation

Vincenzo, on the other hand, completes this rotation task by solely relying on his eyes, without the use of any extra tools. The applet shows how he rotated the triangle 180^(∘) about its fixed vertex.

Notice that a 180^(∘) clockwise rotation and a 180^(∘) counterclockwise rotation produce the same image. Therefore, there is no need to specify direction when rotating a figure by 180^(∘).

Pop Quiz

Practice Rotating Triangles

Rotate triangle ABC around the red point with the specified angle and direction. Use the measuring tool to determine where the vertices of the triangle will end up after the rotation.

Performing random rotations to random triangles

Discussion

Rotating a Point Around the Origin at 90^(∘), 180^(∘), and 270^(∘)

In the coordinate plane, when a point is rotated counterclockwise around the origin at certain angles, its coordinates change in a specific way. This occurs when the angle of rotation is 90^(∘), 180^(∘), or 270^(∘). Try to figure these patterns out using the following applet.

From the diagram, the following rules can be set.

Counterclockwise Rotations Around the Origin
Angle of Rotation	Rule
90^(∘)	(x,y) → (- y,x)
180^(∘)	(x,y) → (- x,- y)
270^(∘)	(x,y) → (y,- x)

Example

A Gateway to a New Realm

The clue Vincenzo and Zosia found reminded them of the word Neptune.

Vincenzo had previously heard stories about this magical place. He had even heard about three ordinary students who had found a door into a mystical library. Could this be another doorway to that place? As they got closer, a puzzle appeared between the columns of the gate.

Graph the figure and its image after a 90^(∘) counterclockwise rotation about the origin on the same coordinate plane.

Answer

Hint

When a point with coordinates (x,y) is rotated 90^(∘) counterclockwise about the origin, its image becomes (- y,x).

Solution

Zosia and Vincenzo want to rotate the figure 90^(∘) counterclockwise around the origin. This can be achieved by rotating each point of the original figure. In this case, four points will be sufficient to determine the position of the image.

When a counterclockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.

Counterclockwise Rotations About the Origin
90^(∘) Rotation	180^(∘) Rotation	270^(∘) Rotation
ccc Preimage & & Image [0.5em] (x,y) & → & (- y,x)	ccc Preimage & & Image [0.5em] (x,y) & → & (- x,- y)	ccc Preimage & & Image [0.5em] (x,y) & → & (y,- x)

Use the coordinate changes shown in the table that correspond to a 90^(∘) counterclockwise rotation about the origin to determine the coordinates of the image of each point. ccc Preimage & & Image [0.3em] (x,y) & → & (- y, x) [0.5em] [-0.5em] A(- 5,- 1) & → & A'(1,- 5) [0.5em] B(- 2,- 3) & → & B'(3,- 2) [0.5em] C(- 5,- 5) & → & C'(5,- 5) [0.5em] D(- 4,- 3) & → & D'(3,- 4) Now plot the image points and connect them. The final figure will be the image of the given figure after the rotation!

Extra

Visualizing the Rotation

The applet shows how the figure is rotated 90^(∘) counterclockwise about the origin. Notice that this rotation maps quadrilateral ABCD onto quadrilateral A'B'C'D'.

Explore

Exploring Dilations

The gate swings open as the figure rotates into place. Zosia takes the first step through the gate, while Vincenzo hesitantly waits to see what happens. Use the sliders to see how Zosia's size changes as she passes through the gate and down the hall.

External credits: Laitche, macrovector

As he watches Zosia's size change with amazement, Vincenzo notices that her image changes relative to a fixed point. How can this point be identified?

Illustration

Dilating a Drawing

Feeling braver, Vincenzo steps through the gate as well. Inside, her and Zosia find an old notebook with a big red dot on one page. Zosia draws a figure on the page and watches as it changes size.

The friends are excited about the magical discovery and eager to comprehend the logic behind it. What other surprises lie ahead?

Discussion

How to Dilate?

As they explore the magical realm, Vincenzo and Zosia come across Dilatius the Dimension Shifter, a wizard who can change the size of objects using dilations. They realize that the notebook they found belongs to the wizard and excitedly ask him to teach them about dilation. Dilatius thrilled to share his knowledge with them.

Concept

Dilation

A dilation is a transformation that changes the size of a figure while keeping its shape the same. This transformation involves enlarging or reducing the figure by a certain length scale factor k from a fixed point O called the center of dilation. For example, the image of every point on a leaf lies on the ray that starts at the center of the dilation and passes through its preimage.

As shown in the diagram, O is the center of the dilation, k is the scale factor, A is the preimage, and A' is the image point. By definition, the scale factor k can also be defined as the ratio of a length in the image to the corresponding length in the preimage.

OA'=k * OA ⇔ k = OA'/OA

When the scale factor is greater than 1, the dilation is called an enlargement because the image is larger than the preimage. When the scale factor is between 0 and 1, the dilation is called a reduction because the image is smaller than the preimage.

Discussion

Coordinate Rule for Dilations

When a point is dilated using a scale factor of k and a center of dilation at the origin, the coordinates of its image are found by multiplying the coordinates of the preimage by k.

(x,y) → ( kx, ky)

The diagram shows how the image changes as the preimage and the scale factor change.

Example

Practicing Dilatius's Teachings

Dilatius is impressed by Vincenzo and Zosia's eagerness to learn. They seem to have picked up the dilation spell using the coordinate rule quickly, so he challenges them to dilate the following triangle.

Draw the image of the triangle after a dilation with center (0,0) and a scale factor of 3.

Answer

Hint

To find the image of a vertex after a dilation with scale factor k, multiply its coordinates by k.

Solution

Zosia and Vincenzo need to dilate the triangle using a scale factor of 3 with respect to the origin. Start by identifying the vertices of the triangle.

When the center of dilation is the origin, each coordinate of the preimage is multiplied by the scale factor k to find the coordinates of the image. ccc Preimage & & Image [0.5em] (x,y)& ⇒ & ( kx, ky) Find the coordinates of the vertices of △ A'B'C' after a dilation with a scale factor k= 3.

Dilation With Scale Factor k=3
Preimage	Multiply by k	Image
A(0,2)	( 3* 0, 3 * 2 )	A'(0,6)
B(3,1)	( 3* 3, 3 * 1 )	B'(9,3)
C(2,- 1)	( 3* 2, 3 * (- 1) )	C'(6,- 3)

Finally, plot the image points and connect them with segments. The new triangle will be the image of the given figure after the dilation!

Checking Our Answer

Visualizing the Dilation

To check the answer, draw rays from the origin through the vertices of the original figure. The vertices of the dilation should lie on those rays.

Notice that OA' is 3 times as long as OA since 3 is the scale factor. This also applies to other side lengths. OA' = 3 * OA OB' = 3 * OB OC' = 3 * OC

Example

Dilatius's Final Lesson

Dilatius teaches Vincenzo and Zosia to use reducio to make things smaller and enlargio to make them bigger. These phrases produce a reduction and an enlargement, respectively. He then quizzes them about the magic behind the drawings in his notebook.

The green square is a dilation of the blue square. Determine whether the dilation is an enlargement or a reduction and the scale factor of the dilation. rcc & Type of Dilation & Scale Factor [1em] A. & Reduction & 1/2 [1em] B. & Enlargement & 3/2 [1em] C. & Reduction & 2/3 [1em] D. & Enlargement & 3

Hint

The scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the original figure.

Solution

There are two types of dilations.

Enlargement: The image is larger than the original figure. An enlargement is the product of a scale factor greater than 1.
Reduction: The image is smaller than the original figure. A reduction is the product of a scale factor between 0 and 1.

In the given coordinate plane, it can be seen that the green square is smaller than the blue square. This means that the dilation is a reduction.

Next, remember that the scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the preimage. Scale Factor= Image Side Length/Preimage Side Length In the graph, AB is 9 units long and the corresponding side A'B' is 6 units long.

Substitute these values into the formula to find the scale factor. Scale Factor= 6/9 = 2/3 The scale factor is equal to 23. Therefore, the green square is a reduction of the blue square with a scale factor of 23. The answer is C.

Checking Our Answer

Visualizing the Dilation

To check the answer, draw rays from the origin through the vertices of the original figure. The vertices of the dilation should lie on those rays.

The square ABCD has a side length of 9 units, while the square A'B'C'D' has a side length of 6 units. Therefore, the scale factor is 69, or 23.

Closure

Comparing Rotations and Dilations

Reflecting on their journey, Vincenzo and Zosia realized that both rotations and dilations preserve the original shape of an object. Consider, for example, a triangle. No matter how much it is rotated or dilated, its image is still a triangle.

However, these transformations do still transform the preimage, almost like a magic trick. Rotations can change the orientation of a figure. Dilations reduce or enlarge a figure while preserving its original shape and angle measures. Since dilations do not preserve side lengths, they are called non-rigid transformations.

Level 1

Level 2

Level 3

Rotations and Dilations

Exercise 2.1

A rotation about the origin maps △ ABC to △ A'B'C'. Which graph shows an angle that could be measured to determine the angle of rotation about the origin?

We know that a rotation about the origin maps △ ABC to △ A'B'C'. We want to determine which of the provided graphs shows an angle that we could measure to find the angle of rotation about the origin. Let's recall how to find an angle of rotation.

Draw a ray from the center of rotation — in this case, the origin — through a vertex in the preimage.
Draw a matching ray from the center of rotation through the vertex's image.
Measure the angle formed by the rays.

With this in mind, let's consider the graphs one by one. Instead of rays, we will consider the dotted segments to determine the angle of rotation.

Graph A

Let's take a look at the first graph.

In the graph we see a segment that connects A and B'. These two points do not correspond to a preimage-image pair because the image of point A is A', not B'. Additionally, the segment does not pass through the origin, which is the center of rotation. Therefore, we cannot use this graph to find the angle of rotation.

Graph B

Now let's apply the same reasoning as we did before to Graph B.

The point B' is not the image of the point A, so we cannot use this graph to determine the angle of rotation. The angle formed by points A, the origin, and B' is not the angle of rotation.

Graph C

Here is another graph to consider.

In this graph, we see a segment that connects B and its image B' and that passes through the origin. This means that the angle formed by points B, the origin, and B' is the angle of rotation. It is 180^(∘) as they form a straight angle. The answer is C.

Graph D

Let's consider our last graph.

Again, the point C' is not the image of the point B, so we cannot use this graph to find the angle of rotation.

The vertices of isosceles right triangle DEF are D(- 5,4), E(- 5,- 3), and F(a,b). The sides DE and DF of this triangle are equal to each other. What are the possible coordinates of the image F' after a 90^(∘) clockwise rotation about the origin?

We know that the isosceles right triangle DEF has vertices D(- 5,4), E(- 5,- 3), and F(a,b). We will start by finding the coordinates of F. Let's graph points D and E.

We can see that the distance between the points is equal to 7 units. Since triangle DEF is isosceles, DE=DF. This means that the length of the second leg DF must also be 7. We can plot F in two different positions, either to the left of D or to the right of D.

We have two possible locations for F. Triangle I & Triangle II F(2,4) & F(- 12,4) Now we can perform the 90^(∘) clockwise rotation of △ DEF. Note that a 90^(∘) clockwise rotation is equivalent to a 270^(∘) counterclockwise rotation. Let's recall the formula for a 270^(∘) counterclockwise rotation about the origin and apply it to each vertex of △ DEF.

Triangle I		Triangle II
(x,y)	(y ,- x)	(x,y)	(y ,- x)
D(- 5,4)	D'(4,5)	D(- 5,4)	D'(4,5)
E(- 5, - 3)	E'(- 3,5)	E(- 5, - 3)	E'(- 3,5)
F(2,4)	F'(4,- 2)	F(- 12,4)	F'(4,12)

Therefore, the image of vertex F after a 90^(∘) clockwise rotation about the origin can be either (4,- 2) or (4,12). Let's draw the possible triangles and perform the rotation.

When a counterclockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.

Counterclockwise Rotations About the Origin
90^(∘) Rotation	180^(∘) Rotation	270^(∘) Rotation
ccc Preimage & & Image [0.5em] (x,y) & → & (- y,x)	ccc Preimage & & Image [0.5em] (x,y) & → & (- x,- y)	ccc Preimage & & Image [0.5em] (x,y) & → & (y,- x)

Davontay draws a triangle with vertices F(3,3), G(7,2), and H(8,- 5). He then rotates the triangle 180^(∘) about the origin. Which transformation creates an image identical to the one Davontay made by rotating the triangle?

Let's start by finding the coordinates of the image of Davontay's triangle after the 180^(∘) rotation about the origin. This rotation changes both the x- and y-coordinates into their opposites. ( x, y) ⟶ ( - x, - y) Let's apply this rule to the vertices of the triangle.

Preimage (x,y)	Image (- x, - y)	Simplify
F( -6, -2)	F'( - 3, - 3 )	F'(- 3,- 3)
G( 7, 2)	G'( - 7 , - 2 )	G'(- 7,- 2)
H( 8, -5)	H'( - 8 , -( -5) )	H'(- 8,5)

The rotated triangle has the following coordinates. F'(- 3,- 3), G'(- 7, - 2), H'(- 8, 5) Now let's draw the image of the triangle after the rotation!

We want to get this image using alternative transformations. First, let's remember how the rotation changed the coordinates of points. ( x, y) ⟶ ( - x, - y) Recall that reflections across the axes change the sign of coordinates, but only one at a time.

Transformation	Effect on Coordinates
Reflection in the x-axis	Changes the y-coordinate to its opposite
Reflection in the y-axis	Changes the x-coordinate to its opposite

If we perform only one of these transformations, we will not get to the desired result. However, if we perform both of them one after the other, we will be able to recreate the rotation! ( x, y) ⇓ c Reflection in thex-axis ( x, - y) ⇓ c Reflection in they-axis ( - x, - y) Performing two reflections leaves us with the same result as performing one 180^(∘) rotation about the origin. Since this works for any point, it will also work for the vertices the triangle!

When we reflect the preimage first in the x-axis and then in the y-axis, we get the same image as the rotation 180^(∘) around the origin.

A square is dilated using a scale factor of 5. The resulting image is then dilated again using a scale factor of 120. What scale factor should be used to dilate the original square to get the final image?

We are told that a square is dilated using a scale factor of 5. After that, the image is dilated using a scale factor of 120. We want to find a scale factor that dilates the original preimage square into the final image. Recall that to find the image of a vertex after a dilation with scale factor k, we multiply its coordinates by k. ccc Preimage & & Image [0.5em] (x,y)& ⇒ & ( kx, ky) This means that if we apply two dilations, we actually apply two scale factors. ccc Preimage & & Image [0.5em] (x,y)& ⇒ & ( k_1* x, k_1* y) ( k_1x, k_1y)& ⇒ & ( k_2 * k_1x, k_2 * k_1 y) This means that the scale factor of the final image will be the product of the first scale factor multiplied by the second scale factor. In our case, the first scale factor is 5 and the second is 120. Let's multiply them!

Therefore, dilating the original square by the scale tfrac14 will result in the same image as the original process.

A line segment has endpoints U(- 8,- 9) and V(- 8,1). Which of the following pairs of points could be the endpoints of the image after a dilation?

We know that a line segment has endpoints U(-8,-9) and V(-8,1). We want to determine which one of the given answers represents a figure that is an image of the given segment after a dilation. A. & U'(- 8, 9) andV'(- 8,- 1) B. & U'(8, 9) and V'(8,1) C. & U'(- 16, - 18) and V'(- 16,2) D. & U'(80, 90) and V'(- 80,10) Recall that when a dilation by a scale factor of k is applied to a point, both of its coordinates are multiplied by the scale factor, k. ccc Preimage & & Image (x,y) & ⇒ & ( kx, ky) To check which of the pairs of points could be the endpoints of the image after a dilation, we need to check if there is a scale factor k that could produce this figure. Recall that the scale factor of dilation is a ratio between the coordinates in the image and the preimage. k=Image Coordinate/Corresponding Preimage Coordinate In other words, to check if a figure is the image of the given segment, we can divide each coordinate of the potential image's endpoints by the corresponding preimage coordinates. If the scale factors from each coordinate are equal, the figure is the image. Let's do it!

	Point U'		Point V'
Option	x-coordinate	y-coordinate	x-coordinate	y-coordinate
A	- 8/- 8=1	9/- 9=-1	-8/- 8=1	- 1/1=-1
B	8/- 8=-1	9/-9=- 1	8/- 8=-1	1/1=1
C	-16/-8=2	-18/-9=2	-16/-8=2	2/1=2
D	80/-8=- 10	90/-9= - 10	- 80/- 8= 10	10/1=10

The scale factor is the same for every coordinate only for the points from option C, so the segment with endpoints U'(- 16,- 18) and V'(- 16,2) is the image of the line segment UV after a dilation. The answer is C.

Diego and Jordan are creating shadows on the wall using a flashlight and their hands. Jordan traces the outline of Diego's shadow on the wall.

Which statements are true? Select all that apply. I. & The type of dilation here is an enlargement. II. & As the distance between Diego and the & flashlight decreases, the shadow on the wall & becomes smaller. III. & The flashlight represents the center of dilation.

If Diego's index finger is 6 inches long and the length of the index finger's shadow is 15 inches, what is the scale factor?

We know that we can use a flashlight to project shadow puppets on a wall. Let's take a look at the visualization.

As we can see, the shadow cast on the wall is larger than our hand. This means that the shadow is the image of our hand after a dilation. This type of dilation is an enlargement because the image is larger than the preimage, which in our case is our hand. Therefore, Statement I is true. ✓ I. & The type of dilation here is an enlargement. The second statement is false because the shadow on the wall becomes larger as the flashlight gets closer to our hand. * II. & As the distance between Diego and the & flashlight decreases, the shadow on the & wall becomes smaller. Remember that if we draw lines connecting corresponding vertices of the image and preimage, the lines will meet at a point called the center of dilation. For shadow puppets, the light rays meet at the flashlight, so the flashlight is the center of dilation. ✓ III. & The flashlight represents the center & of dilation. As a result, Statements I and III are true.

We know that Diego's index finger is 6 inches long. We want to find the scale factor of the dilation when the index finger's shadow is 15 inches long. Remember that the scale factor is the ratio of the length in the image to the corresponding length in the preimage. Scale Factor= Image Length/Corresponding Preimage Length In our case, the measurement of the image is 15 inches and the measurement of the preimage is 6 inches. Let's substitute these values into the formula and simplify.

The scale factor is 52, or 2.5.

Rotations and Dilations

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