DashboardNewsSettingsHelp
Get Premium
Dashboard
Dashboard Sign out
PA
Pre-Algebra View details
2. Rotations and Dilations
Continue to next lesson
Lesson
Exercises
Tests
arrow_back
eCourses /
Pre-Algebra /
Chapter 11
2. 

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.

Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
16 Theory slides
13 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
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.

Rotating a Triangle 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.

Rotation of point A around center P

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.

Points P and A

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

1
Draw PA
expand_more
Using the straightedge, draw the segment connecting the center of rotation P and point A.

Drawing the segment connecting P and A

2
Place the Protractor
expand_more
Place the center of the protractor on P and align it with PA.

Placing the protractor on segment PA so that the rotation is counterclockwise

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.

Placing the protractor so that the rotation is clockwise

3
Mark the Desired Angle
expand_more
Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at 130^(∘).

Making a mark at 130 degrees

4
Draw a Ray
expand_more
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

5
Draw Point A'
expand_more
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.

Points P, A, and A' (image of A under a 130 degree rotation about P)

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.

Trianglewith vertices N, E and O

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.

Triangle NEO, a protractor centered at point O, and a ray starting at O and forming a straight line with side 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.

Triangle NEO and ray ON'

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.

Triangle NEO and a ray starting at O and forming a straight line with side EO

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

Triangle NEO and triangle N'E'O'

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.

Rotations of a points about the origin
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.

External credits: macrovector

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.

  1. Enlargement: The image is larger than the original figure. An enlargement is the product of a scale factor greater than 1.
  2. 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 3.1

On Paulina's treasure map, * marks the spot. She currently standing at point P.

The following transformations will lead Paulina to the treasure, but they are not in the correct order. I. & Reflect in they-axis II. & Rotate90^(∘) counterclockwise about the origin III. & Translate1 unit right and1 unit down IV. & Rotate180^(∘) about the origin Using each transformation exactly once, which of the following orders will guide Paulina to the treasure?

Let's take a look at the given map. It displays a green point representing Paulina's position and a red X indicating the location of the treasure on the coordinate plane.

We want to get to the treasure by applying each of the given transformations only once.

Transformation Rule
I Reflect in the y-axis (x,y) → (- x, y)
II Rotate 90^(∘) counterclockwise about the origin (x,y) → (- y,x)
III Translate 1 unit right and 1 unit down (x,y) → (x+1,y-1)
IV Rotate 180^(∘) about the origin (x,y) → (- x,- y)

We will proceed by trying the order in the options. Our starting point is (- 4,- 2). Let's start with the order in Option A.

The order in Option A led us to the treasure at (3,3). The answer is A. Using a similar table, we can verify that the remaining options do not lead to the treasure.


There are many ways to reach the treasure using these four transformations. Use the applet to apply the given transformations in any order. How many paths to the treasure can you find?


E
Hint & Answer
S
Solution

Rotations and Dilations

Recommended exercises

To get personalized recommended exercises, please login first.
Return to lesson.
Edit Lesson
≥
>
■2
■▢
e▢
7
8
9
×
÷■1​
=
=
√▢​
▢​√
∫▢▢​
4
5
6
+
−
≤
<
log
ln
log▢​
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y