4 Transformations - Mathematics -...

170 Chapter 4

4 Transformations

Mathematical Practices: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

4.1 Translations4.2 Reflections4.3 Rotations4.4 Congruence and Transformations4.5 Dilations4.6 Similarity and Transformations

Revolving Door (p. 195)

Kaleidoscope (p. 196)

Magnification (p. 213)

Chess (p. 179)

Magnification (p. 213)

Kaleidoscope (p 196)

Revolving Door (p 195)

SEE the Big IdeaSEE the Big Idea

Chess (p. 179)

Photo Stickers (p. 211)Photo Stickers (p. 211)

HSCC_GEOM_PE_04.OP.indd 170 2/24/14 10:43 AM

Chapter 4 Pacing GuideChapter Opener/Mathematical Practices

1 Day

Section 1 2 Days

Section 2 1 Day

Section 3 2 Days

Quiz 1 Day

Section 4 1 Day

Section 5 2 Days

Section 6 1 Day

Chapter Review/Chapter Tests

2 Days

Total Chapter 4 13 Days

Year-to-Date 50 Days

HSCC_GEOM_TE_04OP.indd 170HSCC_GEOM_TE_04OP.indd 170 6/5/14 1:52 PM6/5/14 1:52 PM

Dynamic Teaching Tools

Dynamic Assessment & Progress Monitoring Tool

Interactive Whiteboard Lesson Library

Lesson Planning Tool

Dynamic Classroom with Dynamic Investigations

Real-Life STEM Videos






Real-Life STEM Videos

Chapter 4 T-170

Laurie’s Notes

Scaffolding in the ClassroomHow do you solve that?Verbally state the thought processes you use to solve a problem. What questions do you ask yourself? Help the students develop their own set of questions.

Chapter Summary• Students should have a conceptual understanding of transformations from middle school,

where they studied translations, reflections, and rotations. Some may also have been introduced to glide reflections.

• The focus on plane transformations in the Common Core State Standards makes sense in terms of the continuity from middle school to high school. In middle school, the conditions for triangle congruence were informally explored. In high school, “once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.” The criteria for triangle similarity are established through similarity transformations.

• In this chapter, key postulates and theorems relating to rigid motions are presented. Translations, reflections, glide reflections, and rotations are all postulated to be rigid motions. While all could be proven and established as theorems, we have chosen to treat them as postulates in this book.

• Dilations are introduced as nonrigid transformations where the scale factor k results in an enlargement (k > 1) or a reduction (0 < k < 1). The composition of a dilation with rigid motions results in a similarity transformation. In the last lesson, similar figures are defined in terms of similarity transformations.

• The use of dynamic geometry software for the explorations and formal lessons is highly encouraged. This tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students.

Standards Summary

Section Common Core State Standards

4.1 Learning HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.B.6

4.2 LearningHSG-CO.A.2, HSG-CO.A.3, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.B.6, HSG-MG.A.3

4.3 LearningHSG-CO.A.2, HSG-CO.A.3, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.B.6

4.4 Learning HSG-CO.A.5, HSG-CO.B.6

4.5 Learning HSG-CO.A.2, HSG-SRT.A.1a, HSG-SRT.A.1b

4.6 Learning HSG-CO.A.5, HSG-SRT.A.2

Middle School• Draw polygons in the coordinate plane given the vertices, and

find lengths of sides.• Identify congruent figures and similar figures.• Verify the properties of rotations, reflections, and translations.

Algebra 1• Translate, reflect, stretch, and shrink graphs of functions.• Combine transformations of graphs of functions.• Use slope to solve real-life problems.

Geometry• Perform translations, reflections, rotations, dilations, and

compositions of transformations.• Solve real-life problems involving transformations.• Identify lines of symmetry and rotational symmetry.• Describe and perform congruence transformations and

similarity transformations.

COMMON COMMON CORECORE PROGRESSION PROGRESSION

HSCC_GEOM_TE_04OP.indd T-170HSCC_GEOM_TE_04OP.indd T-170 6/5/14 1:52 PM6/5/14 1:52 PM

Laurie’s Notes

T-171 Chapter 4

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyIdentifying Transformations• Remind students that a translation slides a figure; a reflection flips a figure; a rotation turns

a figure. • Of the transformations translation, reflection, rotation, and dilation, the only one that changes

the size of a figure is a dilation.COMMON ERROR Students may confuse reflections and rotations. Remind students that a reflection produces a mirror image of the original figure.

Identifying Similar Figures• Remind students that two figures are similar if and only if corresponding angles are

congruent and the lengths of corresponding sides are proportional.COMMON ERROR Students may forget that congruent figures are also similar figures. Remind students that the ratio of the lengths of corresponding sides in congruent figures is 1:1.

Mathematical PracticesMathematical Practices (continued on page 172)• The eight Mathematical Practices focus attention on how mathematics is learned—process

versus content. Page 172 demonstrates the use of dynamic geometry software as a tool in learning important mathematics.

Questioning in the ClassroomAll students need to participate.When a small group of students continually answers the questions in your classroom, no one else has to think. Try asking a different student whether they agree with the answer given and then follow up with why or why not.

If students need help... If students got it...

Student Journal• Maintaining Mathematical Proficiency

Game Closet at BigIdeasMath.com

Lesson Tutorials Start the next Section

Skills Review Handbook

HSCC_GEOM_TE_04OP.indd T-171HSCC_GEOM_TE_04OP.indd T-171 6/5/14 1:52 PM6/5/14 1:52 PM

Chapter 4 171

171

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyIdentifying Transformations (8.G.A.1)

Example 1 Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure.

a. The blue fi gure b. The red fi gure is a

turns to form mirror image of the

the red fi gure, blue fi gure, so it is

so it is a rotation. a refl ection.

Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure.

1. 2. 3. 4.

Identifying Similar Figures (8.G.A.4)

Example 2 Which rectangle is similar to Rectangle A?

8

41

6

34

Rectangle A

Rectangle BRectangle C

Each figure is a rectangle, so corresponding angles are congruent.

Check to see whether corresponding side lengths are proportional.

Rectangle A and Rectangle B Rectangle A and Rectangle C

Length of A

— Length of B

= 8 —

4 = 2

Width of A —

Width of B =

4 —

1 = 4

Length of A —

Length of C =

8 —

6 =

4 —

3

Width of A —

Width of C =

4 —

3

not proportional proportional

So, Rectangle C is similar to Rectangle A.

Tell whether the two figures are similar. Explain your reasoning.

5.

12

5

7

14

6. 9

12

6

10 158

7.

6

3

510

8. ABSTRACT REASONING Can you draw two squares that are not similar? Explain your reasoning.

Dynamic Solutions available at BigIdeasMath.com

HSCC_GEOM_PE_04.OP.indd 171 5/28/14 3:54 PM

Common Core State Standards

8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations.

8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

ANSWERS1. refl ection

2. rotation

3. dilation

4. translation

5. no; 12

— 14

= 6 —

7 ≠

5 —

7 , The sides are not

proportional.

6. yes; The corresponding angles are

congruent and the corresponding side

lengths are proportional.

7. yes; The corresponding angles are

congruent and the corresponding side

lengths are proportional.

8. no; Squares have four right angles, so

the corresponding angles are always

congruent. Because all four sides are

congruent, the corresponding sides

will always be proportional.

Have students make an Example and Non-Example Chart for the following terms.

• Translation• Reflection• Rotation• Dilation

Vocabulary ReviewVocabulary Revvieew


172 Chapter 4

172 Chapter 4 Transformations

Mathematical Mathematical PracticesPracticesUsing Dynamic Geometry Software

Mathematically profi cient students use appropriate tools strategically, including dynamic geometry software. (MP5)

Monitoring ProgressMonitoring ProgressUse dynamic geometry software to draw the polygon with the given vertices. Use the software to fi nd the side lengths and angle measures of the polygon. Round your answers to the nearest hundredth.

1. A(0, 2), B(3, −1), C(4, 3) 2. A(−2, 1), B(−2, −1), C(3, 2)

3. A(1, 1), B(−3, 1), C(−3, −2), D(1, −2) 4. A(1, 1), B(−3, 1), C(−2, −2), D(2, −2)

5. A(−3, 0), B(0, 3), C(3, 0), D(0, −3) 6. A(0, 0), B(4, 0), C(1, 1), D(0, 3)

Finding Side Lengths and Angle Measures

Use dynamic geometry software to draw a triangle with vertices at A(−2, −1), B(2, 1), and

C(2, −2). Find the side lengths and angle measures of the triangle.

SOLUTIONUsing dynamic geometry software, you can create △ABC, as shown.

−2 −1 0

0

−2

−3

−1

1

2

1 2 3

A B

C

From the display, the side lengths are AB = 4 units, BC = 3 units, and AC = 5 units.

The angle measures, rounded to two decimal places, are m∠A ≈ 36.87°, m∠B = 90°, and m∠C ≈ 53.13°.

Using Dynamic Geometry SoftwareDynamic geometry software allows you to create geometric drawings, including:

• drawing a point

• drawing a line

• drawing a line segment

• drawing an angle

• measuring an angle

• measuring a line segment

• drawing a circle

• drawing an ellipse

• drawing a perpendicular line

• drawing a polygon

• copying and sliding an object

• refl ecting an object in a line

Core Core ConceptConcept

SamplePointsA(−2, 1)B(2, 1)C(2, −2)SegmentsAB = 4BC = 3AC = 5Anglesm∠A = 36.87°m∠B = 90°m∠C = 53.13°

HSCC_GEOM_PE_04.OP.indd 172 2/24/14 10:43 AM

MONITORING PROGRESS ANSWERS

1.

AB ≈ 4.24; BC ≈ 4.12; AC ≈ 4.12;

m∠A ≈ 59.04°; m∠B ≈ 59.04°; m∠C ≈ 61.93°

2.

AB = 2; BC ≈ 5.83; AC ≈ 5.10;

m∠A ≈ 101.31°; m∠B ≈ 59.04°; m∠C ≈ 19.65°

3.

AB = 4; BC = 3; CD = 4; AD = 3;

m∠A = 90°; m∠B = 90°; m∠C = 90°; m∠D = 90°

4.

AB = 4; BC ≈ 3.16; CD = 4;

AD ≈ 3.16; m∠A ≈ 108.43°; m∠B ≈ 71.57°; m∠C ≈ 108.43°; m∠D ≈ 71.57°

5–6. See Additional Answers.

0

1

3

−2

−1

2

4

−1 0 1 2 4 5

A

B

C

0

2

1

−1

−2

0

3

−1 1 2 3 54

B

A

C

0

−1

−3

−4 0

2

−2 −1 2

A

D

B

C

0

1

−2

−1

−3

−4 0

2

−2−3 −1 21

AB

DC

Mathematical PracticesMathematical Practices (continued from page T-171)Laurie’s Notes• Use the Mathematical Practices page to help students develop mathematical habits of mind—

how mathematics can be explored and how mathematics is thought about.• Refer to the Core Concept, and then have students explore the home screen of the software they

will use in this chapter if they have not done so earlier in the year.• In Example 1, make sure students know how to measure lengths and angles.• Throughout this course, you want students to develop the habit of using the dynamic geometry

software as a tool to investigate mathematics and develop understanding of geometric properties and relationships.

• Students should try at least one triangle and one quadrilateral, perhaps Questions 1 and 6.












Section 4.1 T-172

Laurie’s NotesLaurie’s Notes

Overview of Section 4.1Introduction• Students are familiar with translations (slides) from middle school. They learned to translate

figures in the plane and represent a translation using coordinate notation.• The explorations provide an opportunity for students to become familiar with the dynamic

geometry software if they are not already. They need to know how to draw a polygon and translate it by some vector. Students will also verify that a translation preserves the segment length of the preimage for a specific case.

• The formal lesson introduces a key postulate and a key theorem relating to rigid motions. Stating that a translation is a rigid motion is the first postulate of the chapter. The rigid motions, reflection and rotation, are postulated in later lessons. An important and useful theorem presented in this lesson is the Composition Theorem (Thm. 4.1).

• Translations are represented most often in this lesson using coordinate notation, though vector notation is also used. When working with dynamic geometry software, students will need to be familiar with vector notation.

Resources• MP5 Use Appropriate Tools Strategically: Dynamic geometry software, graph paper,

and tracing paper are all tools that will be helpful for students to make sense of translations. Transparencies (overhead projector) and interactive whiteboards are useful tools for teachers.

Common Misconceptions• Students sometimes think that just a few points are translated, such as the endpoints of

segments or vertices of a polygon. Help students to understand that a translation is a function that maps all points of the preimage in the plane to a new location called the image.

Formative Assessment Tips• Always-Sometimes-Never True (AT-ST-NT): This strategy is useful in assessing whether

students over-generalize or under-generalize a particular concept.• MP2 Reason Abstractly and Quantitatively and MP3 Construct Viable Arguments

and Critique the Reasoning of Others: When answering, a student should be asked to justify his or her answer (MP2), and other students listening to the justification should critique the reasoning (MP3).

• AT-ST-NT statements help students practice the habit of checking validity when a statement (conjecture) is made. Are there different cases that need to be checked? Is there a counterexample that would show the conjecture to be false?

• Using AT-ST-NT statements encourages discourse. To develop these statements for a lesson, consider the common errors or misconceptions that students have relating to the goal(s) of the lesson. Allow private think time before students share their thinking with partners or the whole class.

Pacing Suggestion• The formal lesson is long; however, experience with the explorations helps students develop

essential understanding of translations. Students should work through Examples 2–4 more quickly as a result of the explorations.

HSCC_GEOM_TE_0401.indd T-172HSCC_GEOM_TE_0401.indd T-172 6/5/14 1:53 PM6/5/14 1:53 PM

Laurie’s Notes

T-173 Chapter 4

ExplorationMotivate• Ask for four volunteers. Hand them a piece of yarn or rope at least 12 feet long that has been

knotted to form a loop. Have students form the yarn in the shape of a rectangle by holding the vertices.

• Tell students that you are going to give them instructions to move and that on the word “go” they will all move at the same time.

• Examples: “Take two steps to the front of the classroom. Go.”“Take three steps toward the door. Go.”

• Discuss the results of each instruction. In particular, what happened to the rectangle? The rectangle should have remained the same—congruent.

• Repeat with four new students and ask them to form a trapezoid. Tap three of the students on the shoulder and tell them to follow your instructions. The remaining student is to ignore your instructions.

• When you give the first instruction, it is clear that unless all four students move at the same time and in the same direction, the trapezoid is going to change shape!

Exploration 1• This first exploration serves to familiarize students with the translate and copy commands in

the dynamic geometry software. It also uses the language and notation of transformations, both of which should be familiar from middle school.

• Students should observe that the side lengths and angle measures of the copied triangle are the same as the measures in the original triangle.

Exploration 2• Students complete this exploration with paper and pencil. Expect that students recall how to

write the rule for translating an ordered pair using coordinate notation. (The software uses vector notation: <a, b> .)

• None of the sides of △ ABC are vertical or horizontal. Finding the lengths of the sides requires students to use the Distance Formula.

Always-Sometimes-Never True: ”When a segment is translated, the length of the image is congruent to the original segment.” always

Exploration 3Assessing Question: “How can you determine whether a triangle in the coordinate plane is a right triangle?” Find the slopes of the two sides that form the right angle and see whether their product is −1.

Always-Sometimes-Never True: “When an angle is translated, the resulting angle is congruent to the original angle.” always

Communicate Your AnswerCommunicate Your Answer• Neighbor Check: Have students work independently, and then have their neighbors check

their work. Have students discuss any discrepancies.

Connecting to Next Step• Students have now been introduced to translations. In the formal lesson, students will

perform translations using vector notation.


HSG-CO.A.2 Represent transformations in the plane using, e.g., … geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs … .

HSG-CO.A.4 Develop definitions of … translations in terms of angles, … parallel lines, and line segments.

HSG-CO.A.5 Given a geometric figure and a … translation, draw the transformed figure using, e.g., graph paper, … or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HSG-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; …












Section 4.1 173

Section 4.1 Translations 173

Translations4.1

Translating a Triangle in a Coordinate Plane

Work with a partner.

a. Use dynamic geometry software to draw any triangle and label it △ABC.

b. Copy the triangle and translate (or slide) it to form a new fi gure, called an image,

△A′B′C′ (read as “triangle A prime, B prime, C prime”).

c. What is the relationship between the coordinates of the vertices of △ABC and

those of △A′B′C′?d. What do you observe about the side lengths and angle measures of

the two triangles?

0

1

2

3

4

−1

−1

−2

0 1 2 3 4 5 76

A B

C

A′ B′

C′

SamplePointsA(−1, 2)B(3, 2)C(2, −1)SegmentsAB = 4BC = 3.16AC = 4.24Anglesm∠A = 45°m∠B = 71.57°m∠C = 63.43°

Translating a Triangle in a Coordinate Plane


a. The point (x, y) is translated a units horizontally and b units vertically. Write a rule

to determine the coordinates of the image of (x, y).

(x, y) → ( , ) b. Use the rule you wrote in part (a) to translate △ABC 4 units left and 3 units down.

What are the coordinates of the vertices of the image, △A′B′C′?c. Draw △A′B′C′. Are its side lengths the same as those of △ABC ? Justify

your answer.

Comparing Angles of Translations


a. In Exploration 2, is △ABC a right triangle? Justify your answer.

b. In Exploration 2, is △A′B′C′ a right triangle? Justify your answer.

c. Do you think translations always preserve angle measures? Explain your reasoning.

Communicate Your AnswerCommunicate Your Answer 4. How can you translate a fi gure in a coordinate plane?

5. In Exploration 2, translate △A′B′C ′ 3 units right and 4 units up. What are the

coordinates of the vertices of the image, △A″B ″C ″? How are these coordinates

related to the coordinates of the vertices of the original triangle, △ABC ?

Essential QuestionEssential Question How can you translate a fi gure in a

coordinate plane?

COMMON CORE

Learning StandardsHSG-CO.A.2HSG-CO.A.4HSG-CO.A.5HSG-CO.B.6

USING TOOLS STRATEGICALLYTo be profi cient in math, you need to use appropriate tools strategically, including dynamic geometry software.

x

y

4

2

−4

−2

4−2−4

A

B

C

HSCC_GEOM_PE_04.01.indd 173 2/24/14 10:22 AM

ANSWERS1. a. Check students’ work.

b. Check students’ work.

c. The x-values of each of the

three vertices in the image can

be attained by adding the same

amount (positive or negative) to

the corresponding x-values of

the vertices in the original fi gure.

The same is true for the y-values.

d. The side lengths and angle

measures of the original fi gure

are equal to the corresponding

side lengths and angle measures

of the image.

2. a. x + a; y + b

b. A′(−4, 0), B′(0, 2), C′(−1, −6)

c.

yes; Use the Distance Formula to

fi nd the lengths.

3. a. yes; (AB)2 + (AC)2 = (BC)2

b. yes; The side lengths of the

image are the same as the original

fi gure.

c. yes; The image is congruent

to the original fi gure, so the

corresponding angles will be

congruent.

4. Move each vertex the same number

of units left or right, and up or

down. Connect the vertices with a

straightedge.

5. A″(−1, 4), B″(3, 6), C″(2, −2); Each

vertex of the image is 1 unit left and

1 unit up from the corresponding

vertex in the original triangle.

x

y

4

6

−4

−6

4 6−2−4−6

B

C

C′

A

A′B′

HSCC_GEOM_TE_0401.indd 173HSCC_GEOM_TE_0401.indd 173 6/5/14 1:53 PM6/5/14 1:53 PM

174 Chapter 1


What You Will LearnWhat You Will Learn Perform translations.

Perform compositions.

Solve real-life problems involving compositions.

Performing TranslationsA vector is a quantity that has both direction and magnitude, or size, and is

represented in the coordinate plane by an arrow drawn from one point to another.

4.1 Lesson

vector, p. 174initial point, p. 174terminal point, p. 174horizontal component, p. 174vertical component, p. 174component form, p. 174transformation, p. 174image, p. 174preimage, p. 174translation, p. 174rigid motion, p. 176composition of

transformations, p. 176

Core VocabularyCore Vocabullarry

STUDY TIPYou can use prime notation to name an image. For example, if the preimage is point P, then its image is point P′, read as “point P prime.”

Core Core ConceptConceptVectorsThe diagram shows a vector. The initial point, or starting point, of the vector is P, and the

terminal point, or ending point, is Q. The vector

is named PQ , which is read as “vector PQ.” The

horizontal component of PQ is 5, and the vertical component is 3. The component form of a vector

combines the horizontal and vertical components.

So, the component form of PQ is ⟨5, 3⟩.

Identifying Vector Components

In the diagram, name the vector and write its component form.

SOLUTION The vector is JK . To move from the initial point J to the terminal point K, you move

3 units right and 4 units up. So, the component form is ⟨3, 4⟩.

Core Core ConceptConceptTranslationsA translation moves every point of

a fi gure the same distance in the

same direction. More specifi cally,

a translation maps, or moves, the

points P and Q of a plane fi gure along

a vector ⟨a, b⟩ to the points P′ and Q′, so that one of the following

statements is true.

• PP′ = QQ′ and — PP′ � — QQ′ , or

• PP′ = QQ′ and — PP′ and — QQ′ are collinear.

x

y

P(x1, y1)

P′(x1 + a, y1 + b)

Q′(x2 + a, y2 + b)

Q(x2, y2)

A transformation is a function that moves or changes a fi gure in some way to

produce a new fi gure called an image. Another name for the original fi gure is the

preimage. The points on the preimage are the inputs for the transformation, and the

points on the image are the outputs.

P

Q

5 units right

3 unitsup

Translations map lines to parallel lines and segments to parallel segments. For

instance, in the fi gure above, — PQ � — P′Q′ .

J

K


Extra Example 1Name the vector and write its component form.

Q

P

PQ ; ⟨−4, 5⟩

Differentiated Instruction

KinestheticHave students draw and cut out a scalene triangle, place it on the coordinate plane, and trace it in blue. Then ask them to move the triangle 2 units right and 3 units down, and trace it in red. Tell them that the red triangle is the image of the blue triangle after a translation. Have students move their red triangle in a different direction, trace the new image in green, and then describe the translation of the original triangle to the green image.

Teacher ActionsTeacher ActionsLaurie’s Notes• Vectors were introduced when using dynamic geometry software

and are formally defined here. Later you will make the connection between representing translations using vector notation and coordinate notation.

• Big Idea: Note that points on the preimage are inputs and points on the image are outputs. Transformations are functions in the plane.

• Write the Core Concept, Translations, and spend time discussing the two bullets. PP′ = QQ′ means that the length of the two segments is the same. PP′ || QQ′ means that the segments are parallel. All points of PQ are translated the same distance in the same direction.

• Do not become so preoccupied with notation that students fail to see the simplicity of translations.


Section 4.1 175


Writing a Translation Rule

Write a rule for the translation of △ABC to △A′B′C′.

SOLUTIONTo go from A to A′, you move 4 units left and 1 unit up, so you move along the

vector ⟨−4, 1⟩.

So, a rule for the translation is (x, y) → (x − 4, y + 1).x

y

3

42 86

BC

AA′

B′C′

Translating a Figure Using a Vector

The vertices of △ABC are A(0, 3), B(2, 4), and C(1, 0). Translate △ABC using the

vector ⟨5, −1⟩.

SOLUTION

First, graph △ABC. Use ⟨5, −1⟩ to move each

vertex 5 units right and 1 unit down. Label the

image vertices. Draw △A′B′C′. Notice that the

vectors drawn from preimage vertices to image

vertices are parallel.

You can also express translation along the vector ⟨a, b⟩ using a rule, which has the

notation (x, y) → (x + a, y + b).

x

y

2

8

BA

C

B′(7, 3)

A′(5, 2)

C′(6, −1)

Translating a Figure in the Coordinate Plane

Graph quadrilateral ABCD with vertices A(−1, 2), B(−1, 5), C(4, 6), and D(4, 2)

and its image after the translation (x, y) → (x + 3, y − 1).

SOLUTIONGraph quadrilateral ABCD. To fi nd the coordinates of the vertices of the image, add

3 to the x-coordinates and subtract 1 from the y-coordinates of the vertices of the

preimage. Then graph the image, as shown at the left.

(x, y) → (x + 3, y − 1)

A(−1, 2) → A′(2, 1)

B(−1, 5) → B′(2, 4)

C(4, 6) → C′(7, 5)

D(4, 2) → D′(7, 1)

Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com

1. Name the vector and write its component form.

2. The vertices of △LMN are L(2, 2), M(5, 3), and N(9, 1). Translate △LMN using

the vector ⟨−2, 6⟩.

3. In Example 3, write a rule to translate △A′B′C′ back to △ABC.

4. Graph △RST with vertices R(2, 2), S(5, 2), and T(3, 5) and its image after the

translation (x, y) → (x + 1, y + 2).

x

y

4

6

42 6

BC

A DA′

B′C′

D′

K

B


Extra Example 2The vertices of △ ABC are A(0, 3), B(2, 4), and C(1, 0). Translate △ ABC using the vector ⟨−1, −2⟩. A′(−1, 1), B′(1, 2), C′(0, −2)

Extra Example 3Write a rule for the translation of △ ABC to △ A′B′C′.

x

y

2

42 6

C

B

AA′

B′

C′

(x, y) → (x + 2, y − 1)

Extra Example 4Graph quadrilateral ABCD with vertices A(1, −2), B(2, 1), C(4, 1), and D(4, −2) and its image after the translation (x, y) → (x − 1, y + 4).

x

y

4

6

−2

2

D′

C′

A′

B′

B C

DA

English Language Learners

Class ActivityCreate a sheet with two columns. In the first column, show the graphs of four translations. In the second column, show the rules for the four translations—out of order. Have students match each graph with its rule and explain their choices.

Teacher ActionsTeacher ActionsLaurie’s Notes• Examples 2 and 3 make the connection between vector notation and coordinate notation.• Connection: Students will often comment that the translation arrows make the figure look

three-dimensional. The edges of a prism are parallel, so this should be the case.• Alternate Approach: In Example 4, some students will find it easier to plot the vertices of

the image using a strategy of “right 3 and down 1” and then recording the coordinates of the new vertices. The technique shown is to apply the translation rule to find the coordinates of the new vertices, and then those vertices are plotted.

• Monitoring Progress: Use Think-Pair-Share to assess student understanding.


1. BK , ⟨−5, 2⟩2. L′(0, 8), M′(3, 9), N′(7, 7)

3. (x, y) → (x + 4, y − 1)

4. See Additional Answers.


176 Chapter 4


Performing CompositionsA rigid motion is a transformation that preserves length and angle measure. Another

name for a rigid motion is an isometry. A rigid motion maps lines to lines, rays to rays,

and segments to segments.

Performing a Composition

Graph — RS with endpoints R(−8, 5) and S(−6, 8) and its image after the composition.

Translation: (x, y) → (x + 5, y − 2)

Translation: (x, y) → (x − 4, y − 2)

SOLUTION

Step 1 Graph — RS .

Step 2 Translate — RS 5 units right and

2 units down. — R′S′ has endpoints

R′(−3, 3) and S′(−1, 6).

Step 3 Translate — R′S′ 4 units left and

2 units down. — R″S″ has endpoints

R″(−7, 1) and S″(−5, 4).

TheoremTheoremTheorem 4.1 Composition TheoremThe composition of two (or more) rigid motions is a rigid motion.

Proof Ex. 35, p. 180

P

QP′

Q′

P″

Q″

translation 1

com

posit

ion

translation 2

x

y

4

2

8

6

−2−4−6−8

R(−8, 5)

S(−6, 8)

R″(−7, 1)

S″(−5, 4)

R′(−3, 3)

S′(−1, 6)

Because a translation is a rigid motion, and a rigid motion preserves length and angle

measure, the following statements are true for the translation shown.

• DE = D′E′, EF = E′F′, FD = F′D′

• m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′

When two or more transformations are combined to form a single transformation, the

result is a composition of transformations.

D F

E

E′

F′D′

Postulate 4.1 Translation PostulateA translation is a rigid motion.

PostulatePostulate

The theorem above is important because

it states that no matter how many rigid

motions you perform, lengths and angle

measures will be preserved in the fi nal

image. For instance, the composition of

two or more translations is a translation,

as shown.


Extra Example 5Graph RS with endpoints R(−8, 5) and S(−6, 8). Graph its image after the composition.Translation: (x, y) → (x − 1, y + 4)Translation: (x, y) → (x + 4, y − 6)

x

y

4

6

8

2

−2−4−6−8

S″(−3, 6)

R″(−5, 3)

S(−6, 8)

R(−8, 5)

Teacher ActionsTeacher ActionsLaurie’s Notes• The definition of a rigid motion (isometry) is simple to state and it

says a lot! Remind students that a transformation moves points in the plane, and when the length and angle measure are preserved, it is called a rigid motion. Recall the Motivate when only three of the four students followed directions. Length and angle measure were not preserved and, hence, the transformation was not rigid. Students will learn that there are only three rigid motions in the plane. The Translation Postulate (Post. 4.1) identifies the first rigid motion.

• Connection: Students have done composition of functions in algebra. If f(x) = x2 and g(x) = x + 2, then g(f(x)) = x2 + 2.

• The statement of the Composition Theorem (Thm. 4.1) is simple, yet its importance is enormous. As figures in the plane are mapped onto new figures, regardless of the number of times, segment lengths and angle measures are preserved. The final image is congruent to the preimage.

• Alternate Approach: Students should have graph paper or dynamic geometry software available for Example 5.


Section 4.1 177


Solving Real-Life Problems

Modeling with Mathematics

You are designing a favicon for a

golf website. In an image-editing

program, you move the red rectangle

2 units left and 3 units down. Then

you move the red rectangle 1 unit

right and 1 unit up. Rewrite the

composition as a single translation.

SOLUTION

1. Understand the Problem You are

given two translations. You need to

rewrite the result of the composition

of the two translations as a

single translation.

2. Make a Plan You can choose an arbitrary point (x, y) in the red rectangle and

determine the horizontal and vertical shift in the coordinates of the point after both

translations. This tells you how much you need to shift each coordinate to map the

original fi gure to the fi nal image.

3. Solve the Problem Let A(x, y) be an arbitrary point in the red rectangle. After the

fi rst translation, the coordinates of its image are

A′(x − 2, y − 3).

The second translation maps A′(x − 2, y − 3) to

A″(x − 2 + 1, y − 3 + 1) = A″(x − 1, y − 2).

The composition of translations uses the original point (x, y) as the input and

returns the point (x − 1, y − 2) as the output.

So, the single translation rule for the composition is (x, y) → (x − 1, y − 2).

4. Look Back Check that the rule is correct by testing a point. For instance, (10, 12)

is a point in the red rectangle. Apply the two translations to (10, 12).

(10, 12) → (8, 9) → (9, 10)Does the fi nal result match the rule you found in Step 3?

(10, 12) → (10 − 1, 12 − 2) = (9, 10) ✓


5. Graph — TU with endpoints T(1, 2) and U(4, 6) and its image after the composition.


Translation: (x, y) → (x − 4, y + 5)

6. Graph — VW with endpoints V(−6, −4) and W(−3, 1) and its image after the

composition.

Translation: (x, y) → (x + 3, y + 1)


7. In Example 6, you move the gray square 2 units right and 3 units up. Then you

move the gray square 1 unit left and 1 unit down. Rewrite the composition as a

single transformation.

x

y

4

2

8

6

12

10

14

42 86 1210 14


Extra Example 6Another graphic artist is designing an alternate icon for the one in the graph in Example 6. She moves the red rectangle 3 units right and 1 unit down. Then she moves the red rectangle 1 unit left and 4 units up. Rewrite the composition as a single transformation. (x, y) → (x + 2, y + 3)

Teacher ActionsTeacher ActionsLaurie’s Notes “Does anyone know what a favicon is? Explain.” Answers will vary.

• FYI: A favicon is a custom icon associated with a webpage or website that appears next to the http address in the URL window of a browser. It will also occupy open tabs in tab-enabled browsers, and it will display next to bookmarked links. The term combines the words “favorites“ and “icon.“

• The problem posed is not difficult to solve with mental math. The point is not that you can solve the problem mentally. The practice

you want to provide students is the opportunity to read a problem, make sense of the context, and translate words into symbols. This is a building block for future problems. I ask a volunteer to read Steps 1 and 2 in the problem-solving plan.

ClosureClosure• Draw a triangle in Quadrant I. Have students identify a translation

that would map the triangle entirely to Quadrant III. Write the translation using coordinate notation and vector notation.


5.

6.

7. (x, y) → (x + 1, y + 2)

x

y

4

6

2

−2

2 4−2−4

8

T″

U″

U

T

x

2

−2

−4

−6

−2−6−8

y

V

V′

W

W′


178 Chapter 4



1. VOCABULARY Name the preimage and image of the transformation △ABC → △A′B′C ′.

2. COMPLETE THE SENTENCE A ______ moves every point of a fi gure the same distance in the

same direction.

Exercises4.1

Vocabulary and Core Concept CheckVocabulary and Core Concept Check

In Exercises 3 and 4, name the vector and write its component form. (See Example 1.)

3.

C

D

4. S

T

In Exercises 5–8, the vertices of △DEF are D(2, 5), E(6, 3), and F(4, 0). Translate △DEF using the given vector. Graph △DEF and its image. (See Example 2.)

5. ⟨6, 0⟩ 6. ⟨5, −1⟩

7. ⟨−3, −7⟩ 8. ⟨−2, −4⟩

In Exercises 9 and 10, fi nd the component form of the vector that translates P(−3, 6) to P′.

9. P′(0, 1) 10. P′(−4, 8)

In Exercises 11 and 12, write a rule for the translation of △LMN to △L′M′N ′. (See Example 3.)

11.

x

y4

−2

6−2−4NL

MN′L′

M′

12.

NLN′L′

x

y1

−5

31−3−7 MM′

In Exercises 13–16, use the translation.

(x, y) → (x − 8, y + 4)

13. What is the image of A(2, 6)?

14. What is the image of B(−1, 5)?

15. What is the preimage of C ′(−3, −10)?

16. What is the preimage of D′(4, −3)?

In Exercises 17–20, graph △PQR with vertices P(−2, 3), Q(1, 2), and R(3, −1) and its image after the translation. (See Example 4.)

17. (x, y) → (x + 4, y + 6)

18. (x, y) → (x + 9, y − 2)

19. (x, y) → (x − 2, y − 5)

20. (x, y) → (x − 1, y + 3)

In Exercises 21 and 22, graph △XYZ with vertices X(2, 4), Y(6, 0), and Z(7, 2) and its image after the composition. (See Example 5.)

21. Translation: (x, y) → (x + 12, y + 4)


22. Translation: (x, y) → (x − 6, y)


Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics

HSCC_GEOM_PE_04.01.indd 178 5/28/14 3:54 PM

9. ⟨3, −5⟩ 10. ⟨−1, 2⟩ 11. (x, y) → (x − 5, y + 2)

12. (x, y) → (x + 3, y + 1)

13. A′(−6, 10)

14. B′(−9, 9)

15. C(5, −14)

16. D(12, −7)

17.


x

4

6

8

2 4 6−2

y

P′ Q′

R′

P Q

R

ANSWERS 1. △ABC is the preimage, and △A′B′C′

is the image.

2. translation

3. CD , ⟨7, −3⟩ 4. ST , ⟨−2, −4⟩ 5.

6.

7.

8.

x

4

6

2

−2

4 8 12−2

y

DED′

E′

F F′

x

4

6

2

−2

4 8 12−2

y

D

E

D′

E′

FF′

x

8

4

−4

−8

4 8−4

y

D′E′

F′

DE

F

x

4

2

−2

−4

2 4 6 8−2

y

D′

E′

F′

D

E

F

Assignment Guide and Homework Check

ASSIGNMENT

Basic: 1, 2, 3–25 odd, 26, 32, 39, 43–50

Average: 1, 2, 4, 8–24 even, 25, 26, 32–38 even, 39, 43–50

Advanced: 1, 2, 6, 10, 16, 24–26, 28–32, 34–36, 39–50

HOMEWORK CHECK

Basic: 3, 5, 15, 21, 32

Average: 4, 16, 18, 26, 34

Advanced: 16, 26, 28, 35, 41










Section 4.1 179


In Exercises 23 and 24, describe the composition of translations.

23.

x

y4

2

−2

42−2−4

A

C B

A″

C″ B″

A′

B′C′

24.

x

y

3

−2

5−1

D

G F

E

D″

G″

E″

F″

D′

F′G′

E′

25. ERROR ANALYSIS Describe and correct the error in

graphing the image of quadrilateral EFGH after the

translation (x, y) → (x − 1, y − 2).

E

HG

F

G′

F′E′

H′

x

y

3

5

1

31 5 9

✗

26. MODELING WITH MATHEMATICS In chess, the

knight (the piece shaped like a horse) moves in an

L pattern. The board shows two consecutive moves

of a black knight during a game. Write a composition

of translations for the moves. Then rewrite the

composition as a single translation that moves

the knight from its original position to its ending

position. (See Example 6.)

27. PROBLEM SOLVING You are studying an amoeba

through a microscope. Suppose the amoeba moves on

a grid-indexed microscope slide in a straight line from

square B3 to square G7.

7654321

A B C D E F G H

8X

a. Describe the translation.

b. The side length of each grid square is

2 millimeters. How far does the amoeba travel?

c. The amoeba moves from square B3 to square G7

in 24.5 seconds. What is its speed in millimeters

per second?

28. MATHEMATICAL CONNECTIONS Translation A maps

(x, y) to (x + n, y + t). Translation B maps (x, y) to

(x + s, y + m).

a. Translate a point using Translation A, followed by

Translation B. Write an algebraic rule for the fi nal

image of the point after this composition.

b. Translate a point using Translation B, followed by

Translation A. Write an algebraic rule for the fi nal

image of the point after this composition.

c. Compare the rules you wrote for parts (a) and

(b). Does it matter which translation you do fi rst?

Explain your reasoning.

MATHEMATICAL CONNECTIONS In Exercises 29 and 30, a translation maps the blue fi gure to the red fi gure. Find the value of each variable.

29.

s 2tr°

162° 100°

3w°

10

8

30.

4c − 6

b + 6a°

55°

20

14


ANSWERS23. translation: (x, y) → (x + 5, y + 1),

translation: (x, y) → (x − 5, y − 5)

24. translation: (x, y) → (x + 6, y − 4),

translation: (x, y) → (x − 6, y)

25. The quadrilateral should have been

translated left and down;

26. translation: (x, y) → (x + 2, y − 1),

translation: (x, y) → (x + 1, y − 2);

(x, y) → (x + 3, y − 3)

27. a. The amoeba moves right 5 and

down 4.

b. about 12.8 mm

c. about 0.52 mm/sec

28. a. (x, y) → (x + n + s, y + t + m)

b. (x, y) → (x + s + n, y + m + t)

c. no; Each image will end up in the

same place.

29. r = 100, s = 8, t = 5, w = 54

30. a = 35, b = 14, c = 5

x

2

−2

2

y

4

G′H′

F′

F

HE

GE′


180 Chapter 4


Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyTell whether the fi gure can be folded in half so that one side matches the other. (Skills Review Handbook)

43. 44. 45. 46.

Simplify the expression. (Skills Review Handbook)

47. −(−x) 48. −(x + 3) 49. x − (12 − 5x) 50. x − (−2x + 4)

Reviewing what you learned in previous grades and lessons

31. USING STRUCTURE Quadrilateral DEFG has vertices

D(−1, 2), E(−2, 0), F(−1, −1), and G(1, 3). A

translation maps quadrilateral DEFG to quadrilateral

D′E′F′G′. The image of D is D′(−2, −2). What are

the coordinates of E′, F′, and G′?

32. HOW DO YOU SEE IT? Which two fi gures represent

a translation? Describe the translation.

14

7

8

9

6

5

2

3

33. REASONING The translation (x, y) → (x + m, y + n)

maps — PQ to

— P′Q′ . Write a rule for the translation of

— P′Q′ to

— PQ . Explain your reasoning.

34. DRAWING CONCLUSIONS The vertices of a rectangle

are Q(2, −3), R(2, 4), S(5, 4), and T(5, −3).

a. Translate rectangle QRST 3 units left and 3 units

down to produce rectangle Q′R′S′T ′. Find the

area of rectangle QRST and the area of

rectangle Q′R′S′T ′.

b. Compare the areas. Make a conjecture about

the areas of a preimage and its image after

a translation.

35. PROVING A THEOREM Prove the Composition

Theorem (Theorem 4.1).

36. PROVING A THEOREM Use properties of translations

to prove each theorem.

a. Corresponding Angles Theorem (Theorem 3.1)

b. Corresponding Angles Converse (Theorem 3.5)

37. WRITING Explain how to use translations to draw

a rectangular prism.

38. MATHEMATICAL CONNECTIONS The vector

PQ = ⟨4, 1⟩ describes the translation of A(−1, w)

onto A′(2x + 1, 4) and B(8y − 1, 1) onto B′(3, 3z).

Find the values of w, x, y, and z.

39. MAKING AN ARGUMENT A translation maps — GH to —G′H′ . Your friend claims that if you draw segments

connecting G to G′ and H to H′, then the resulting

quadrilateral is a parallelogram. Is your friend

correct? Explain your reasoning.

40. THOUGHT PROVOKING You are a graphic designer

for a company that manufactures fl oor tiles. Design a

fl oor tile in a coordinate plane. Then use translations

to show how the tiles cover an entire fl oor. Describe

the translations that map the original tile to four

other tiles.

41. REASONING The vertices of △ABC are A(2, 2),

B(4, 2), and C(3, 4). Graph the image of △ABCafter the transformation (x, y) → (x + y, y). Is this

transformation a translation? Explain your reasoning.

42. PROOF —MN is perpendicular to lineℓ. —M′N′ is the

translation of —MN 2 units to the left. Prove that

—M′N′ is perpendicular toℓ.


ANSWERS31. E′(−3, −4), F′(−2, −5), G′(0, −1)

32. fi gures 5 and 7; To go from fi gure 5

to fi gure 7, you move 4 units right

and 8 units up.

33. (x, y) → (x − m, y − n); You must go

back the same number of units in the

opposite direction.



Resources by Chapter • Practice A and Practice B• Puzzle Time

Resources by Chapter• Enrichment and Extension• Cumulative Review

Student Journal • Practice

Start the next Section

Differentiating the LessonSkills Review Handbook

Mini-Assessment

1. The vertices of △ ABC are A(0, 3), B(2, 4), and C(1, 0). Translate △ ABC using the vector ⟨−4, 3⟩. A′(−4, 6), B′(−2, 7), C′(−3, 3)

2. Graph RS with endpoints R(−4, 2) and S(2, −2) and its image after the translation (x, y) → (x + 5, y − 1).

x

y

2

−4

−2

4 62−2−4

R′(1, 1)

S′(7, −3)

R(−4, 2)

S(2, −2)

3. Graph quadrilateral ABCD with vertices A(1, −2), B(2, 1), C(4, 1), and D(4, −2) and its image after the composition.



x

y

−2

2−2 C″

D″A″

B″

C

DA

B2

4. A gardener transplants a small bush, moving it 3 feet east and 10 feet south of its original position. She realizes that the plant is not getting enough sunlight, so she moves it 8 feet west and 2 feet north. Using addition to represent moves east or north and subtraction to represent moves west or south, write the composition as a single translation.(x, y) → (x − 5, y − 8)

HSCC_GEOM_TE_0401.indd 180HSCC_GEOM_TE_0401.indd 180 6/6/14 11:45 AM6/6/14 11:45 AM











Section 4.2 T-180

Laurie’s Notes

Overview of Section 4.2Introduction• Students worked with reflections of polygons in middle school. In algebra, they explored the

transformation of y = x2 to y = −x2 as a reflection in the x-axis.• The explorations provide an opportunity for students to become familiar with using dynamic

geometry software to perform reflections. Inductively, students will develop rules for reflections in the x- and y-axes. They also explore preservation of segment length using the Distance Formula.

• The formal lesson includes the definitions of reflection and glide reflection. The coordinate rules for reflections in the x- and y-axes and in the lines y = x and y = −x are also developed. Once the postulate stating that a reflection is a rigid motion is given, glide reflections are introduced as the composition of a translation followed by a reflection; hence, it too is a rigid motion by the Composition Theorem (Thm. 4.1).

• Students may forget that the line of reflection is the perpendicular bisector of the segment connecting a point on the preimage and its corresponding point on the image, when the point is not on the line of reflection. Recalling this property will be helpful in later lessons.

Resources• MP5 Use Appropriate Tools Strategically: In addition to dynamic geometry software,

graph paper, and tracing paper mirrors, there are other tools that will be helpful for students to make sense of reflections. Exploration 1 is designed to use a reflective device.

Formative Assessment Tips• Fact-First Questioning: This is a higher-order questioning technique that goes beyond

asking straight recall questions. Instead, this strategy allows you to assess students’ growing understanding of a concept or skill.

• First, state a fact. Then ask students why, or how, or to explain. Student thinking is activated, and you gain insight into the depth of students’ conceptual understanding.

• Example: State the fact, “All squares are rectangles.” Then ask, “Why is this true?”

Applications• There are particular occupations that come to mind in which the ability to work with

reflections is absolutely essential. Dentists see a reflection of your teeth as they do their work. A driver backing up and using his or her side or rearview mirrors is working with a reflected image.

Pacing Suggestion• Once students have worked through the explorations, you could assign Exercises 3–12 on

page 186. You may consider omitting Example 1 and Monitoring Progress Questions 1–4.


Laurie’s Notes

T-181 Chapter 4

ExplorationMotivate• 60-Second Write: Tell students that they have 60 seconds to make a list of words that can

be reflected in a vertical or horizontal line and the result is the same word or a new word.• Example: When the word MOM is reflected in a vertical line, it is still MOM. When it is

reflected in a horizontal line, it becomes WOW.• Share word lists after 60 seconds. The list should contain letters that have line symmetry

themselves: A, B, C, D, E, H, I, K, M, O, T, V, W, X, and Y.• Explain that reflections and line symmetry are studied in this lesson.

Exploration 1• If you have reflective devices, have students complete the construction of the reflected

triangle. “Which point of the preimage is closest to line m?” Answers vary. “Which point of the image is closest to line m?” Answers vary, but for every student the two answers are related.

Exploration 2• This exploration serves to familiarize students with the reflect command in the dynamic

geometry software. In using the software, students will learn that they must select a polygon, in this case the triangle, and also the line in which the reflection will be done.

• Encourage students to draw a scalene triangle so that the correspondence of sides will be easier to see following the reflection.

• Students should observe that the side lengths and angle measures of the reflected triangle are the same as the measures in the original triangle. They should also observe a relationship between the coordinates of corresponding vertices. Using lattice points (points at the intersections of grid lines) for vertices will help students recognize a pattern.

• The generalizations students should recognize are the coordinate representations of (x, y) when a point is reflected in the y-axis and in the x-axis.

Always-Sometimes-Never True: “When an ordered pair is reflected in the x-axis, the y-coordinate of the image is negative.” sometimes true; When the y-coordinate is positive, it will be negative when reflected in the x-axis. When the y-coordinate is negative, it will be positive when reflected in the x-axis. When the y-coordinate is 0, it will be 0 when reflected in the x-axis.

• Extension: Use software to reflect a triangle in any horizontal or vertical line. Ask students to develop a rule for reflecting (x, y) in the line y = x or y = −x.

Communicate Your AnswerCommunicate Your Answer• Expect students to refer to reflections in the x- or y-axis in Question 3.

Connecting to Next Step• If students complete both explorations, skip Example 1 in the formal lesson and check for

understanding with Monitoring Progress Questions 1–4.


HSG-CO.A.2 Represent transformations in the plane using, e.g., … geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. …

HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the … reflections that carry it onto itself.

HSG-CO.A.4 Develop definitions of … reflections, … in terms of angles, … perpendicular lines, parallel lines, and line segments.

HSG-CO.A.5 Given a geometric figure and a … reflection, … draw the transformed figure using, e.g., graph paper, … or geometry software. Specify a sequence of transformations that will carry a given figure onto another.


HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).












Section 4.2 181

Section 4.2 Refl ections 181

Refl ections4.2

Refl ecting a Triangle Using a Refl ective Device

Work with a partner. Use a straightedge to draw any triangle on paper. Label

it △ABC.

a. Use the straightedge to draw a line that does not pass through the triangle.

Label it m.

b. Place a refl ective device on line m.

c. Use the refl ective device to plot the images of the vertices of △ABC. Label the

images of vertices A, B, and C as A′, B′, and C′, respectively.

d. Use a straightedge to draw △A′B′C′ by connecting the vertices.

Refl ecting a Triangle in a Coordinate Plane

Work with a partner. Use dynamic geometry software to draw any triangle and label

it △ABC.

a. Refl ect △ABC in the y-axis to form △A′B′C′.b. What is the relationship between the coordinates of the vertices of △ABC and

those of △A′B′C′?c. What do you observe about the side lengths and angle measures of the two triangles?

d. Refl ect △ABC in the x-axis to form △A′B′C′. Then repeat parts (b) and (c).

0

1

2

3

4

−1−2−3

−1

0 1 2 4

A

C

B

A′

C′

B′

3

1

2

Communicate Your AnswerCommunicate Your Answer 3. How can you refl ect a fi gure in a coordinate plane?

Essential QuestionEssential Question How can you refl ect a fi gure in a

coordinate plane?COMMON CORE

Learning StandardsHSG-CO.A.2HSG-CO.A.3HSG-CO.A.4HSG-CO.A.5HSG-CO.B.6HSG-MG.A.3

LOOKING FOR STRUCTURE

To be profi cient in math, you need to look closely to discern a pattern or structure.

SamplePointsA(−3, 3)B(−2, −1)C(−1, 4)SegmentsAB = 4.12BC = 5.10AC = 2.24Anglesm∠A = 102.53°m∠B = 25.35°m∠C = 52.13°

HSCC_GEOM_PE_04.02.indd 181 2/24/14 10:23 AM 3. If a fi gure is refl ected in the y-axis, then

each pair of corresponding vertices will have

the same y-value and opposite x-values. If

a fi gure is refl ected in the x-axis, then each

pair of corresponding vertices will have the

same x-value and opposite y-values.



c. Sample answer:

d. Sample answer:

2. a. Check students’ work.

b. Each vertex of △A′B′C′ has the

same y-value as its corresponding

vertex of △ABC. The x-value

of each vertex of △A′B′C′ is

the opposite of the x-value of its

corresponding vertex of △ABC.

c. The corresponding sides and

corresponding angles are congruent.

d. Sample answer:

Each vertex of △A′B′C′ has the

same x-value as its corresponding

vertex of △ABC. The y-value of

each vertex of △A′B′C′ is the

opposite of the x-value of its

corresponding vertex of △ABC;

The corresponding sides and

corresponding angles are

congruent.

A

B B′

A′

C C′

m

A

B B′

A′

C C′

m

0

2

−2

−1

−4

−3

−4 0

4

3

1

−3−5 −1 32 4 51

B

A′

C′

C

A

B′


182 Chapter 4


4.2 Lesson What You Will LearnWhat You Will Learn Perform refl ections.

Perform glide refl ections.

Identify lines of symmetry.

Solve real-life problems involving refl ections.

Performing Refl ections

Refl ecting in Horizontal and Vertical Lines

Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the

refl ection described.

a. In the line n: x = 3 b. In the line m: y = 1

SOLUTION

a. Point A is 2 units left of line n, so its

refl ection A′ is 2 units right of line nat (5, 3). Also, B′ is 2 units left of

line n at (1, 2), and C′ is 1 unit right

of line n at (4, 1).

AB

n

Cx

y4

2

42 6

A′

C′B′

b. Point A is 2 units above line m, so

A′ is 2 units below line m at (1, −1).

Also, B′ is 1 unit below line m at

(5, 0). Because point C is on line m,

you know that C = C′.

AB

m

x

y4

2

6

A′

B′C

C′


Graph △ABC from Example 1 and its image after a refl ection in the given line.

1. x = 4 2. x = −3

3. y = 2 4. y = −1

refl ection, p. 182line of refl ection, p. 182glide refl ection, p. 184line symmetry, p. 185line of symmetry, p. 185


Core Core ConceptConceptRefl ections

A refl ection is a transformation that uses a line like a mirror to refl ect a fi gure.

The mirror line is called the line of refl ection.

A refl ection in a line m maps every point

P in the plane to a point P′, so that for

each point one of the following properties

is true.

• If P is not on m, then m is the

perpendicular bisector of — PP′ , or

• If P is on m, then P = P′.

P

m

point P not on m

P′

P

m

point P on m

P′


Extra Example 1Graph △ ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection described.

a. In the line n: x = −1

x

y

4

2

42−2−4−6

C′

A′

n

B′

C

AB

b. In the line m: y = 3

x

y

4

6

2

4 62

C′

A′m

B′

C

A B


VisualHave students hold a pocket mirror on the lines of reflection in Examples 1–3, and ask them to describe what they see.

Teacher ActionsTeacher ActionsLaurie’s Notes• Write the Core Concept. Distinguish the two cases related to the two bullets.• Teaching Tip: Draw two points on scrap paper. Crease the paper to form the line of reflection

for the points. Draw the segment that connects the two points. Use this model to highlight the property of the line of reflection being the perpendicular bisector of the segment joining the two points.

• Students have little difficulty reflecting a point in a horizontal (x = k) or vertical line (y = k). Their spatial skills may make them less able to correctly reflect in a diagonal line.

MONITORING PROGRESS ANSWERS 1.


3.

4.

−2

A′A

C′C

B′ B

x = 4

x

2

2 6

y

A C′BB′

CA′x2 4

y

y = 2

A

C

B

C′

A′B′

x

2

−2

−4

2 4 6−2

y

y = −1


Section 4.2 183


Refl ecting in the Line y = −x

Graph — FG from Example 2 and its image after a refl ection in the line y = −x.

SOLUTIONUse the coordinate rule for refl ecting in the line

y = −x to fi nd the coordinates of the endpoints

of the image. Then graph — FG and its image.

(a, b) → (−b, −a)

F(−1, 2) → F′(−2, 1)

G(1, 2) → G′(−2, −1)


The vertices of △JKL are J(1, 3), K(4, 4), and L(3, 1).

5. Graph △JKL and its image after a refl ection in the x-axis.

6. Graph △JKL and its image after a refl ection in the y-axis.

7. Graph △JKL and its image after a refl ection in the line y = x.

8. Graph △JKL and its image after a refl ection in the line y = −x.

9. In Example 3, verify that — FF′ is perpendicular to y = −x.

Refl ecting in the Line y = x

Graph — FG with endpoints F(−1, 2) and G(1, 2) and its image after a refl ection in the

line y = x.

SOLUTIONThe slope of y = x is 1. The segment from F to

its image, — FF′ , is perpendicular to the line of

refl ection y = x, so the slope of — FF′ will be −1

(because 1(−1) = −1). From F, move 1.5 units

right and 1.5 units down to y = x. From that point,

move 1.5 units right and 1.5 units down to

locate F′(2, −1).

The slope of — GG′ will also be −1. From G, move

0.5 unit right and 0.5 unit down to y = x. Then move

0.5 unit right and 0.5 unit down to locate G′(2, 1).

You can use coordinate rules to fi nd the images of points refl ected in four special lines.

REMEMBERThe product of the slopes of perpendicular lines is −1.

Core Core ConceptConceptCoordinate Rules for Refl ections• If (a, b) is refl ected in the x-axis, then its image is the point (a, −b).

• If (a, b) is refl ected in the y-axis, then its image is the point (−a, b).

• If (a, b) is refl ected in the line y = x, then its image is the point (b, a).

• If (a, b) is refl ected in the line y = −x, then its image is the point (−b, −a).

x

y4

−2

4−2

F G

F′

G′

y = x

F′

G′x

y

−2

2

F G

y = −x


Extra Example 2Graph AB with endpoints A(3, −1) and B(3, 2) and its image after a reflection in the line y = x.

x

y

4

2

−2

42−2

A′ B′

A

B

y = x

Extra Example 3Graph AB with endpoints A(3, −1) and B(3, 2) and its image after a reflection in the line y = −x.

x

y

2

−4

−2

42−2

A′B′

A

B

y = −x

Teacher ActionsTeacher ActionsLaurie’s Notes• One strategy students use to reflect in the line y = x is to rotate their graph paper so that the

line of reflection is horizontal. It is easier for many students to then reflect the endpoints of FG in a vertical direction. Work through the Example 2 solution that involves the slopes of perpendicular lines.

• Before stating the Core Concept that summarizes the rules for four common reflections, try the formative assessment tip, Fact-First Question.

Fact-First Question: “The reflection of (x, y) in the line y = x is (y, x). Why are the coordinates interchanged, or reversed, when a point is reflected in the line y = x?” Listen for explanations involving the slope between (x, y) and (y, x) being −1.

• Think-Pair-Share: Have students work Example 3 and then compare work with their partners.


6.

7.


x

y

4

2

−4

−2

4 62−2

K

L

L′

J

K′J′

x

y

6

2

−2

42−2−4

K′ K

L

J

L′

J′

x

y4

2

42

J′

L′J K′

K

L

y = x


184 Chapter 4


Performing Glide Refl ections

Because a refl ection is a rigid motion, and a rigid motion preserves length and angle

measure, the following statements are true for the refl ection shown.

• DE = D′E′, EF = E′F′, FD = F′D′

• m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′

Because a refl ection is a rigid motion, the Composition Theorem (Theorem 4.1)

guarantees that any composition of refl ections and translations is a rigid motion.

A glide refl ection is a transformation involving a

translation followed by a refl ection in which every

point P is mapped to a point P ″ by the following steps.

Step 1 First, a translation maps P to P′.

Step 2 Then, a refl ection in a line k parallel to the

direction of the translation maps P′ to P ″.

Performing a Glide Refl ection

Graph △ABC with vertices A(3, 2), B(6, 3), and C(7, 1) and its image after the

glide refl ection.

Translation: (x, y) → (x − 12, y)

Refl ection: in the x-axis

SOLUTIONBegin by graphing △ABC. Then graph △A′B′C′ after a translation 12 units left.

Finally, graph △A″B″C″ after a refl ection in the x-axis.

x

y

2

−2

42 86−2−4−6−8−10−12

A(3, 2)

B(6, 3)

C(7, 1)A′(−9, 2)

A″(−9, −2)

B′(−6, 3)

B″(−6, −3)

C′(−5, 1)

C″(−5, −1)


10. WHAT IF? In Example 4, △ABC is translated 4 units down and then refl ected in

the y-axis. Graph △ABC and its image after the glide refl ection.

11. In Example 4, describe a glide refl ection from △A″B″C ″ to △ABC.

STUDY TIPThe line of refl ection must be parallel to the direction of the translation to be a glide refl ection.

PostulatePostulatePostulate 4.2 Refl ection PostulateA refl ection is a rigid motion.

D F

mE E′

F′ D′

P

P′

P″

Q′ Q″

Q

k


Extra Example 4Graph △ ABC with vertices A(3, 2), B(6, 3), and C(7, 1) and its image after the glide reflection.Translation: (x, y) → (x, y − 6)Reflection: in the y-axis

x

y

4

−4

4

A′(3, −4)A″(−3, −4)

B″(−6, −3)

C″(−7, −5)

B′(6, −3)

C′(7, −5)

A(3, 2)B(6, 3)

C(7, 1)

Teacher ActionsTeacher ActionsLaurie’s Notes• Write the Reflection Postulate (Post. 4.2) and ask students to interpret

what it means. When rigid motions or isometries are referred to, students should be thinking about segment lengths and angle measures being preserved.

• Interpret the postulate for △DEF reflected in line m.• Teaching Tip: Show an image (consisting of translations,

reflections, and glide reflections) of the footprints of a person walking a straight line. Ask students to describe the transformation(s) they

see. They will often say that there are translations, left foot to left foot and right foot to right foot. In trying to describe the reflection, they often say that one foot has to be slid ahead and then reflected. Now is the time to define glide reflection!

Turn and Talk: “Is a glide reflection a rigid motion? Explain.” yes; Listen for students to reference the Composition Theorem (Thm. 4.1).

COMMON ERROR Students often forget that the line of reflection must be parallel to the direction of the translation.


KinestheticAnother way to find a line of symmetry is to trace the figure on tracing paper, and then try to fold the paper so that one half of the figure matches up with the other half. When this happens, the fold crease is a line of symmetry. Have students try this method with the figures in Example 5.


11. translation: (x, y) → (x + 12, y),

refl ection: in the x-axis

x

y

2

−2

4 62−2−4−6

AB

C

A′A″

C ″

B″B′

C′


Section 4.2 185


Identifying Lines of SymmetryA fi gure in the plane has line symmetry when the fi gure can be mapped onto itself by

a refl ection in a line. This line of refl ection is a line of symmetry, such as line m at the

left. A fi gure can have more than one line of symmetry.

Identifying Lines of Symmetry

How many lines of symmetry does each hexagon have?

a. b. c.

SOLUTION

a. b. c.


Determine the number of lines of symmetry for the fi gure.

12. 13. 14.

15. Draw a hexagon with no lines of symmetry.


Finding a Minimum Distance

You are going to buy books. Your friend

is going to buy CDs. Where should you

park to minimize the distance you

both will walk?

SOLUTIONRefl ect B in line m to obtain B′. Then

draw — AB′ . Label the intersection of

— AB′ and m as C. Because AB′ is the shortest

distance between A and B′ and BC = B′C,

park at point C to minimize the combined

distance, AC + BC, you both have to walk.


16. Look back at Example 6. Answer the question by using a refl ection of point A

instead of point B.

m

Bm

A

B

C m

A

B′


Extra Example 5How many lines of symmetry does the triangle have?

3

Extra Example 6Use the picture in Example 6. You are going to the music store. Your friend is going to buy craft supplies. Where should you park to minimize the distance you both will walk?

B C

D m

A

C′

Let point C be in front of the Arts and Crafts store, on the same line as AB. You should park at point D.

Teacher ActionsTeacher ActionsLaurie’s Notes• Line symmetry should be familiar to students from middle school

geometry.• Alternate Approach: If you are confident of students’

knowledge of line symmetry, skip Example 5 and Monitoring Progress Questions 12–15 and instead ask students to work with partners to draw hexagons with the following lines of symmetry: 0, 1, 2, 3, 4, 5, and 6. Note: It is not possible to draw a hexagon with exactly 4 or 5 lines of symmetry.

• MP1 Make Sense of Problems and Persevere in Solving Them: Example 6 is a classic problem that can be posed using

a variety of different contexts. (See Exercise 27.) Consider posing the problem for students to explore with partners in class, or for homework. In either scenario, the example can be done with or without dynamic geometry software.

Closure Fact-First Questioning: “The reflection of △CAT is △C′A′T′, and the two triangles have the same perimeter. Why should the triangles have the same perimeter?” Reflection is a rigid motion that preserves length (and angles), so the perimeter of △CAT is the same length as the perimeter of △C′A′T′.

MONITORING PROGRESS ANSWERS 12. 2

13. 5

14. 1

15. Sample answer:

16. Refl ect A in line m to obtain A′. Then

draw — BA′ . Label the intersection of — BA′ and m as C. Because BA′ is the

shortest distance between B and A′ and AC = A′C, park at point C.

mC

B A

A′


186 Chapter 4


Exercises4.2 Dynamic Solutions available at BigIdeasMath.com

1. VOCABULARY A glide refl ection is a combination of which two transformations?

2. WHICH ONE DOESN’T BELONG? Which transformation does not belong with the other three? Explain

your reasoning.


In Exercises 3–6, determine whether the coordinate plane shows a refl ection in the x-axis, y-axis, or neither.

3. 4.

x

y2

−4

−6

4−4 A B

C

DE

F

x

y4

−4

−2

4−4 A

B

C

D

E

F

5. 6.

x

y4

2

42−2−4AB

C

D

E

F

x

y4

2

−4

−2

4A

B

CD

E

F

In Exercises 7–12, graph △JKL and its image after a refl ection in the given line. (See Example 1.)

7. J(2, −4), K(3, 7), L(6, −1); x-axis

8. J(5, 3), K(1, −2), L(−3, 4); y-axis

9. J(2, −1), K(4, −5), L(3, 1); x = −1

10. J(1, −1), K(3, 0), L(0, −4); x = 2

11. J(2, 4), K(−4, −2), L(−1, 0); y = 1

12. J(3, −5), K(4, −1), L(0, −3); y = −3

In Exercises 13–16, graph the polygon and its image after a refl ection in the given line. (See Examples 2 and 3.)

13. y = x 14. y = x

x

y

2

−4

−2

4 6

AB

C

x

y4

−2

4−2A

BC

D

15. y = −x 16. y = −x

x

y4

2

−4

2

A

BC

D

x

y4

2

−4

−2

4 6−2

A B

C


x

y

4

6

2

2−2

x

y

2

−2

42 x

y

2

−2

−2−4 x

y

2

−2−4


ANSWERS1. translation and refl ection

2.

It is a translation, and the other three

are refl ections.

3. y-axis

4. neither

5. neither

6. x-axis

7.

8.

x

y

4

2

−4

−2

42

x

y

4

6

2

−4

−6

−2

42

J′

J

L′

L

K

K′

x

y4

2

4−4

JJ′L′

K′ K

L


ASSIGNMENT

Basic: 1–4, 5–25 odd, 26, 27, 34, 40–49

Average: 1, 2, 4–32 even, 33, 34, 39, 40–49

Advanced: 1, 2, 10, 12, 16, 20–26 even, 27–29, 32–34, 37–49

HOMEWORK CHECK

Basic: 7, 15, 17, 23, 34

Average: 12, 16, 20, 22, 34

Advanced: 12, 16, 20, 32, 34

9.

10.

x

y

−2

−4

4−2−6

L′

J′

K′

L

J

Kx = −1

x

y

−2

4

K′ KJ

J′

L′L−4

x = 2

11.


x

y

4

2

−2

2−4

K′

K J′

L′

L

J

y = 1










Section 4.2 187


In Exercises 17–20, graph △RST with vertices R(4, 1), S(7, 3), and T(6, 4) and its image after the glide refl ection. (See Example 4.)

17. Translation: (x, y) → (x, y − 1)

Refl ection: in the y-axis

18. Translation: (x, y) → (x − 3, y)

Refl ection: in the line y = −1

19. Translation: (x, y) → (x, y + 4)

Refl ection: in the line x = 3


Refl ection: in the line y = x

In Exercises 21–24, determine the number of lines of symmetry for the fi gure. (See Example 5.)

21. 22.

23. 24.

25. USING STRUCTURE Identify the line symmetry

(if any) of each word.

a. LOOK

b. MOM

c. OX

d. DAD


describing the transformation.

x

y

2

−2

42 86−2−4−6−8

BA

B″ B′

A′A″

— AB to — A″B″ is a glide refl ection.✗

27. MODELING WITH MATHEMATICS You park at some

point K on line n. You deliver a pizza to House H,

go back to your car, and deliver a pizza to House J.

Assuming that you can cut across both lawns, how

can you determine the parking location K that

minimizes the distance HK + KJ ? (See Example 6.)

n

H J

28. ATTENDING TO PRECISION Use the numbers and

symbols to create the glide refl ection resulting in the

image shown.

x

y

4

6

2

−4

−2

4 6 82−2−4A(3, 2)

C(2, −4)

B(−1, 1)

A″(5, 6)

B″(4, 2)

C″(−1, 5)

Translation: (x, y) → ( , ) Refl ection: in y = x

x −+y

21 3

In Exercises 29–32, fi nd point C on the x-axis so AC + BC is a minimum.

29. A(1, 4), B(6, 1)

30. A(4, −5), B(12, 3)

31. A(−8, 4), B(−1, 3)

32. A(−1, 7), B(5, −4)

33. MATHEMATICAL CONNECTIONS The line y = 3x + 2

is refl ected in the line y = −1. What is the equation of

the image?

HSCC_GEOM_PE_04.02.indd 187 2/26/14 9:24 AM 26. The line of refl ection has to be parallel to

the direction of the translation for it to be a

guide refl ection; translation

(x, y) → (x + 2, y + 3), refl ection: in the

y-axis

27. Refl ect H in line n to obtain H′. Then draw — JH′ . Label the intersection of JH′ and n

as K. Because JH′ is the shortest distance

between J and H′ and HK = H′K, park at

point K.

28. x + 3, y + 3

29. C(5, 0)

30. C(9, 0)

31. C(−4, 0)

32. C ( 31 —

11 , 0 )

33. y = −3x − 4

ANSWERS17.

18.

19.

20.

21. 1

22. 4

23. 0

24. 5

25. a. none

b.

c.

d. none

x

y

2

4

−2 2 4 6−4−6R′

T″

R″S″ R S′

S

TT′

x

y

2

−2

−4

−6

4

2 4 6

T′

T″

R″

S″

S′

R′ R

ST

y = −1

x

y

2

4

6

42 6

x = 3

T′T″

S″

R″

S′

S

R

TR′

x

y

2

4

6

42 6 8

y = x S′

T′

S″

T″R″

SR′

T

R

8

MOM

OX


188 Chapter 4


34. HOW DO YOU SEE IT? Use Figure A.

x

y

Figure A

x

y

x

y

Figure 1 Figure 2

x

y

x

y

Figure 3 Figure 4

a. Which fi gure is a refl ection of Figure A in the

line x = a? Explain.

b. Which fi gure is a refl ection of Figure A in the

line y = b? Explain.

c. Which fi gure is a refl ection of Figure A in the

line y = x? Explain.

d. Is there a fi gure that represents a glide refl ection?


35. CONSTRUCTION Follow these steps to construct a

refl ection of △ABC in line m. Use a compass

and straightedge.

Step 1 Draw △ABC and line m.

Step 2 Use one compass setting

to fi nd two points that are

equidistant from A on line

m. Use the same compass

setting to fi nd a point on

the other side of m that is

the same distance from

these two points. Label

that point as A′.

Step 3 Repeat Step 2 to fi nd points B′ and C′. Draw △A′B′C′.

36. USING TOOLS Use a refl ective device to verify your

construction in Exercise 35.

37. MATHEMATICAL CONNECTIONS Refl ect △MNQ in

the line y = −2x.

x

y4

−3

1−5

y = −2x

Q

M

N

38. THOUGHT PROVOKING Is the composition of a

translation and a refl ection commutative? (In other

words, do you obtain the same image regardless of

the order in which you perform the transformations?)

Justify your answer.

39. MATHEMATICAL CONNECTIONS Point B′(1, 4) is the

image of B(3, 2) after a refl ection in line c. Write an

equation for line c.

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyUse the diagram to fi nd the angle measure. (Section 1.5)

40. m∠AOC 41. m∠AOD

42. m∠BOE 43. m∠AOE

44. m∠COD 45. m∠EOD

46. m∠COE 47. m∠AOB

48. m∠COB 49. m∠BOD


9090

8010070

1106012050

130

40140

30150

20160

10 170

0 180

10080

11070 12060 13050 14040 15030

1602017010

1800

BOA

E

DC

A

C

B

m


ANSWERS 34–38. See Additional Answers.

39. y = x + 1

40. 60°

41. 130°

42. 20°

43. 160°

44. 70°

45. 30°

46. 100°

47. 180°

48. 120°

49. 50°

Mini-Assessment

△ ABC has vertices A(1, 3), B(4, 1), and C(1, 1).

1. Graph △ ABC and its image after a reflection in the line y = x.

x

y

4

2

42

A′

B′

C′

A

BC

y = x

2. Graph △ ABC and its image after the glide reflection.

Translation: (x, y) → (x − 4, y) Reflection: in the x-axis

x

y4

2

−4

42

A′

A″

B″C″

B′C′

A

BC

3. How many lines of symmetry does the figure have?

1


Resources by Chapter• Practice A and Practice B• Puzzle Time
















Section 4.3 T-188

Laurie’s Notes

Overview of Section 4.3Introduction• Students worked with rotation of polygons in middle school; however, it is unlikely that they

developed coordinate rules for the benchmark rotations.• The explorations provide an opportunity for students to become familiar with using dynamic

geometry software to perform rotations. Inductively, students will develop rules for rotations of 90° and 180° about the origin.

• The formal lesson presents the definition of a rotation and then students use a protractor and compass to draw a rotation. The coordinate rules for rotations about benchmark angles are developed, and this can be a muddy point for students. The rules make sense when looking at an example, but students find them hard to remember. Encourage students to focus on a single point such as (3, 1) and try to visualize the image.

• Big Idea: Students will see that the order in which transformations are performed generally does matter. The Composition Theorem (Thm. 4.1) states that the result of composing two or more rigid motions will be a rigid motion. That is different from the question of the order in which the compositions are done.

Resources• MP5 Use Appropriate Tools Strategically: Dynamic geometry software, graph paper,

and tracing paper are all tools that will be helpful for students to make sense of rotations. Transparencies (overhead projector) and interactive whiteboards are useful tools for teachers.

Teaching Strategy• Students can generally visualize and sketch the translation or reflection of a figure. This is not

true for rotations. The rotation is the most difficult transformation for students to visualize and to sketch. The difficulty is that all points in the preimage turn through the same number of degrees but they do not move the same distance.

• MP4 Model with Mathematics and MP5: To help students visualize the rotation, it is very helpful for them to have tracing paper. For demonstration purposes, it is helpful to have a transparency or a projecting device that allows students to see the fixed point (center of rotation) and the movement of the figure from the initial state (preimage) to the final state (image).

• Use the Motivate to help students visualize rotations.

Pacing Suggestion• Take time for students to work through all three explorations, and then begin the formal

lesson. Alternatively, you may integrate the explorations at point of use in the formal lesson: Do Exploration 1 followed by Example 1. Do Explorations 2 and 3 and then complete the formal lesson.


Laurie’s Notes

T-189 Chapter 4

ExplorationMotivate• A windshield wiper blade is a useful model in this lesson. The next time you change blades,

keep a used blade for a model in teaching this lesson! • Hold the blade at one end of the wiper (point A). Without saying anything, pretend to use the

blade for wiping a windshield (rotating about point A). “Do all points on the blade travel the same distance?” no “Are there any points that are not moving?“ yes; point A “Is there anything that is true about all points on the blade?” yes; They all turn the same number of degrees.

• Now tape the blade to a meterstick and explain, “The blade on a car is attached to an arm.” Model the movement of the blade now and ask questions similar to before. The difference now is that point A is no longer the center of the rotation, so it is moving.

Exploration 1• This first exploration serves to familiarize students with the rotate command in the dynamic

geometry software. In using the software, students will learn that they must select a polygon, a point of rotation, and the number of degrees the figure will be rotated.

• Encourage students to draw a scalene triangle so that the correspondence of sides will be easier to see after the rotation.

• Students should observe that the side lengths and angle measures of the rotated triangle are the same as the measures in the original triangle. They should also observe a relationship between the coordinates of corresponding vertices. Using lattice points (points at intersections of grid lines) for vertices will help students recognize a pattern.

Exploration 2• Students should use the results of Exploration 1 to write the rule for a rotation of 90°

counterclockwise about the origin. If the pattern is not obvious, then suggest that students try additional triangles in Exploration 1.

• None of the sides of △ ABC are vertical or horizontal. Finding the lengths of the sides requires students to use the Distance Formula.

Exploration 3• A rotation of 180° can be investigated using the software. Students should observe that it is

not necessary to specify clockwise or counterclockwise when the angle measure is 180°.

Communicate Your AnswerCommunicate Your Answer• Expect students to mention needing a center of rotation and an angle measure.• Popsicle Sticks: Select a student to explain Question 5.

Connecting to Next Step• Now that students have developed some spatial abilities by using the software, move to the

formal lesson, where a protractor and compass are used to construct a rotation.


HSG-CO.A.2 Represent transformations in the plane using, e.g., … geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs… .

HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations … that carry it onto itself.

HSG-CO.A.4 Develop definitions of rotations, … in terms of angles, circles, … and line segments.

HSG-CO.A.5 Given a geometric figure and a rotation, … draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.



Technology for the Teacher

Computer

Calculator

Online Ancillaries

Smartboard

Video











Section 4.3 189

Technology for the Teacher

Computer

Calculator

Online Ancillaries

Smartboard

Video

Section 4.3 Rotations 189

Rotations4.3

Rotating a Triangle in a Coordinate Plane



b. Rotate the triangle 90° counterclockwise about the origin to form △A′B′C′.c. What is the relationship between the coordinates of the vertices of △ABC and

those of △A′B′C′?d. What do you observe about the side lengths

and angle measures of the two triangles?



a. The point (x, y) is rotated 90° counterclockwise about the origin. Write a rule to

determine the coordinates of the image of (x, y).

b. Use the rule you wrote in part (a) to rotate △ABC 90° counterclockwise about

the origin. What are the coordinates of the vertices of the image, △A′B′C′?c. Draw △A′B′C′. Are its side lengths the same as those of △ABC? Justify

your answer.



a. The point (x, y) is rotated 180° counterclockwise about the origin. Write a rule to

determine the coordinates of the image of (x, y). Explain how you found the rule.

b. Use the rule you wrote in part (a) to rotate △ABC (from Exploration 2) 180°counterclockwise about the origin. What are the coordinates of the vertices of the

image, △A′B′C′?

Communicate Your AnswerCommunicate Your Answer 4. How can you rotate a fi gure in a coordinate plane?

5. In Exploration 3, rotate △A′B′C′ 180° counterclockwise about the origin.

What are the coordinates of the vertices of the image, △A″B″C″? How are

these coordinates related to the coordinates of the vertices of the original

triangle, △ABC?

CONSTRUCTING VIABLE ARGUMENTS

To be profi cient in math, you need to use previously established results in constructing arguments.

Essential QuestionEssential Question How can you rotate a fi gure in a

coordinate plane?COMMON CORE

Learning StandardsHSG-CO.A.2HSG-CO.A.3HSG-CO.A.4HSG-CO.A.5HSG-CO.B.6

x

y5

1

−3

−5

51−1−3−5

A

B

C

0

1

2

3

4

−1−2−3

−1

0 1 2 4

A

D

C

B

A′

C′B′

3

2

3

SamplePointsA(1, 3)B(4, 3)C(4, 1)D(0, 0)SegmentsAB = 3BC = 2AC = 3.61Anglesm∠A = 33.69°m∠B = 90°m∠C = 56.31°




c. The x-value of each vertex of

△A′B′C′ is the opposite of the

y-value of its corresponding

vertex in △ABC. The y-value of

each vertex of △A′B′C′ is equal

to the x-value of its corresponding

vertex in △ABC.

d. The side lengths and angle

measures of the original fi gure

are equal to the corresponding

side lengths and angle measures

of the image. For example,

AB = A′B′ and m∠A = m∠A′. 2. a. (x, y) → (−y, x)

b. A′(–3, 0), B′(−5, 4), C′(3, 3)

c.

yes; Use the Distance Formula to

fi nd the lengths.

3. a. −x; −y; When a point is rotated

180°, the x-value and y-value

of the image are the opposite of

the x-value and y-value of the

original point.

b. A′(0, −3), B′(−4, −5), C′(−3, 3)

4. Sample answer: Put your pencil on

the origin and rotate the graph the

given number of degrees. Record

the coordinates of the image in

this orientation. Then return the

coordinate plane to its original

orientation, and draw the image using

the coordinates you recorded.

5. A″(0, 3), B″(4, 5), C″(3, −3); The

coordinates of each vertex are the

same as the corresponding vertex of

the original triangle.

x

y

4

6

−4

−6

−2

4 62−2−4−6

B

C

AC′

B′

A′


190 Chapter 4


4.3 Lesson What You Will LearnWhat You Will Learn Perform rotations.

Perform compositions with rotations.

Identify rotational symmetry.

Performing Rotations

The fi gure above shows a 40° counterclockwise rotation. Rotations can be clockwise

or counterclockwise. In this chapter, all rotations are counterclockwise unless

otherwise noted.

Drawing a Rotation

Draw a 120° rotation of △ABC about point P.

SOLUTION

Step 1 Draw a segment from P to A.

P

A

BC

Step 2 Draw a ray to form a 120° angle

with — PA .

P

A

BCP

C9090

80100

70110

60 120

50 130

40 140

30 150

20 160

10 170

0 180

10080

11070

12060

13050

14040

15030 16020 17010 1800

Step 3 Draw A′ so that PA′ = PA.

P

A

A′ BC120°

Step 4 Repeat Steps 1–3 for each vertex.

Draw △A′B′C′.

P

A

A′

B′

C′B

C

rotation, p. 190center of rotation, p. 190angle of rotation, p. 190rotational symmetry, p. 193center of symmetry, p. 193


Core Core ConceptConceptRotationsA rotation is a transformation in which a fi gure is turned about a fi xed point

called the center of rotation. Rays drawn from the center of rotation to a point

and its image form the angle of rotation.

A rotation about a point P through an angle

of x° maps every point Q in the plane to

a point Q′ so that one of the following

properties is true.

• If Q is not the center of rotation P,

then QP = Q′P and m∠QPQ′ = x°, or

• If Q is the center of rotation P, then

Q = Q′.

R

Q

P

Q′

R′

40°angle ofrotation

center ofrotation

P

A

BC

clockwise

counterclockwise

Direction of rotation


Extra Example 1Use the diagram in Example 1. Draw a 60° rotation of △ ABC about point P.

P

AA′

B′

C′BC

Teacher ActionsTeacher ActionsLaurie’s Notes• Write the Core Concept. Note how the angle of rotation is defined. Again, differentiate between

the two bullets.• Teaching Tip: Use a transparency at the overhead or document camera, or use an electronic

model that allows the center of rotation to be visible as the rotation is occurring. “If you want to perform a rotation, what do you need to know?” the figure to be rotated, the center of rotation, and how many degrees the figure is rotated

• Mention the convention that rotations in this book will be counterclockwise unless otherwise noted.• Go through the steps necessary for drawing a rotation.

Turn and Talk: “Is it necessary to find the image of all three vertices to rotate a triangle? Explain.” Discuss as a class.


Visual AidStudents often confuse the meaning of the terms clockwise and counterclockwise. Tell them they can look at the movement of the hands of a clock for a visual reminder of the meaning of clockwise.


Section 4.3 191


Rotating a Figure in the Coordinate Plane

Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, −3), and U(2, −1) and

its image after a 270° rotation about the origin.

SOLUTIONUse the coordinate rule for a 270° rotation to

fi nd the coordinates of the vertices of the image.

Then graph quadrilateral RSTU and its image.

(a, b) → (b, −a)

R(3, 1) → R′(1, −3)

S(5, 1) → S′(1, −5)

T(5, −3) → T′(−3, −5)

U(2, −1) → U′(−1, −2)


1. Trace △DEF and point P. Then draw a 50° rotation of △DEF about point P.

PFD

E

2. Graph △JKL with vertices J(3, 0), K(4, 3), and L(6, 0) and its image after

a 90° rotation about the origin.

You can rotate a fi gure more than 180°. The diagram

shows rotations of point A 130°, 220°, and 310° about the origin. Notice that point A and its images

all lie on the same circle. A rotation of 360° maps

a fi gure onto itself.

You can use coordinate rules to fi nd the coordinates

of a point after a rotation of 90°, 180°, or 270° about the origin.

USING ROTATIONSYou can rotate a fi gure more than 360°. The effect, however, is the same as rotating the fi gure by the angle minus 360°.

Core Core ConceptConceptCoordinate Rules for Rotations about the OriginWhen a point (a, b) is rotated counterclockwise

about the origin, the following are true.

• For a rotation of 90°, (a, b) → (−b, a).

• For a rotation of 180°, (a, b) → (−a, −b).

• For a rotation of 270°, (a, b) → (b, −a).

x

y

AA′

A″

A‴

130°

220°

310°

x

y

90°180°

270°

(a, b)(−b, a)

(−a, −b)(b, −a)

x

y2

−6

6−2−4

R S

T

UU′R′

S′T′


Extra Example 2Graph △ ABC with vertices A(3, 1), B(3, 4), and C(1, 1) and its image after a 180° rotation about the origin.

x

y4

2

−4

−2

−2 42A′

B′

C′A

B

C

Teacher ActionsTeacher ActionsLaurie’s Notes “Have you heard of angle measures greater than 360°? Explain.” Students should quickly mention contexts such as skateboarding and snowboarding where multiples of 180° are common.

• MP8 Look For and Express Regularity in Repeated Reasoning: Use dynamic geometry software or a transparency with grid paper to model rotations of 90°, 180°, and 270°. Rotating just one point (which is not near y = x or y = −x) is all that is necessary. For example, using point A(3, 1), the 90° rotation is (−1, 3), the 180° rotation is (−3, −1), and the 270° rotation

is (1, −3). This simple activity will help convince students of the rules that follow.

• Write the coordinate rules for benchmark rotations about the origin. Confirm that the rules work for the example with A(3, 1).

• Whiteboarding: Pose Example 2 and give partners time to work the example. Circulate and ask advancing questions as needed to assist students in making progress.


1.

2.

E

E′

D′

F′

D PF

x

y

4

2

6

4 62−2

K′L′

J′K

J L


192 Chapter 4


Performing Compositions with Rotations

Because a rotation is a rigid motion, and a rigid motion preserves length and angle

measure, the following statements are true for the rotation shown.

• DE = D′E′, EF = E′F′, FD = F′D′

• m∠D = m∠D′, m∠E = m∠E′, m∠F = m∠F′

Because a rotation is a rigid motion, the Composition Theorem (Theorem 4.1)

guarantees that compositions of rotations and other rigid motions, such as translations

and refl ections, are rigid motions.

Performing a Composition

Graph — RS with endpoints R(1, −3) and S(2, −6) and its image after the composition.


Rotation: 90° about the origin

SOLUTION

Step 1 Graph — RS .

Step 2 Refl ect — RS in the y-axis. — R′S′ has endpoints

R′(−1, −3) and S′(−2, −6).

Step 3 Rotate — R′S′ 90° about the

origin. — R″S″ has endpoints

R″(3, −1) and S″(6, −2).


3. Graph — RS from Example 3. Perform the rotation fi rst, followed by the refl ection.

Does the order of the transformations matter? Explain.

4. WHAT IF? In Example 3, — RS is refl ected in the x-axis and rotated 180° about

the origin. Graph — RS and its image after the composition.

5. Graph — AB with endpoints A(−4, 4) and B(−1, 7) and its image after

the composition.



6. Graph △TUV with vertices T(1, 2), U(3, 5), and V(6, 3) and its image after

the composition.



COMMON ERRORUnless you are told otherwise, perform the transformations in the order given.

PostulatePostulatePostulate 4.3 Rotation PostulateA rotation is a rigid motion.

D

F

E

E′

F′ D′

x

y

−6

8−2−4

R(1, −3)

S(2, −6)

R′(−1, −3)

S′(−2, −6)

R″(3, −1)

S″(6, −2)


Extra Example 3Graph — RS with endpoints R(1, −3) and S(2, −6) and its image after the composition.Rotation: 180° about the originReflection: in the y-axis

x

y

4

6

2

−4

−6

−2

42−2−4

R′(−1, 3)

S′(−2, 6)

R″(1, 3)

S″(2, 6)

R(1, −3)

S(2, −6)

Teacher ActionsTeacher ActionsLaurie’s Notes• Write the Rotation Postulate (Post. 4.3) and ask students to interpret what it means. When rigid

motions or isometries are referred to, students should be thinking about segment lengths and angle measures being preserved.

• Interpret the postulate for △DEF rotated about point P.• Alternate Approach: Students could do the composition using software.

Always-Sometimes-Never True: “The order in which you perform two transformations does not matter.” Give partners time to consider the validity of this statement. They think of it as a question of commutativity. The statement is sometimes true for special cases (i.e., the preimage is a point on the line of reflection), but in general it is not true. Monitoring Progress Question 3 explores this.


3.

yes; The image is in Quadrant I, not

Quadrant IV.

4.


x

y2

−4

−6

−2

4 62−2−4−6

S′

R′R″

S″

R

S

x

y

4

−8

8

4 8−4−8

S′

S

R′

R″

S″

R


Section 4.3 193


Identifying Rotational SymmetryA fi gure in the plane has rotational symmetry when the fi gure can be mapped

onto itself by a rotation of 180° or less about the center of the fi gure. This point

is the center of symmetry. Note that the rotation can be either clockwise or

counterclockwise.

For example, the fi gure below has rotational symmetry, because a rotation of either

90° or 180° maps the fi gure onto itself (although a rotation of 45° does not).

0° 45°90°

180°

The fi gure above also has point symmetry, which is 180° rotational symmetry.

Identifying Rotational Symmetry

Does the fi gure have rotational symmetry? If so, describe any rotations that map the

fi gure onto itself.

a. parallelogram b. regular octagon c. trapezoid

SOLUTION

a. The parallelogram has rotational symmetry.

The center is the intersection of the diagonals.

A 180° rotation about the center maps the

parallelogram onto itself.

b. The regular octagon has rotational symmetry.

The center is the intersection of the diagonals.

Rotations of 45°, 90°, 135°, or 180° about the

center all map the octagon onto itself.

c. The trapezoid does not have rotational

symmetry because no rotation of 180° or

less maps the trapezoid onto itself.


Determine whether the fi gure has rotational symmetry. If so, describe any rotations that map the fi gure onto itself.

7. rhombus 8. octagon 9. right triangle


Extra Example 4Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself.

a.

yes, 120°

b.

no

c.

yes, 180°

Teacher ActionsTeacher ActionsLaurie’s Notes• Rotational symmetry should be familiar to students from middle school geometry.• Teaching Tip: Provide tracing paper for students to use.• Extension: Draw a hexagon with 60° rotational symmetry.

ClosureClosure• Muddiest Point: Ask students to identify, aloud or on a paper to be collected, the muddiest

point(s) about the lesson. What was difficult to understand?

MONITORING PROGRESS ANSWERS 7. yes; The center is the intersection

of the diagonals. A rotation of 180°

about the center maps the rhombus

onto itself.

8. yes; The center is the intersection

of the diagonals. Rotations of 90°

and 180° about the center map the

octagon onto itself.

9. no


KinestheticHave students draw and cut out a rectangle and a parallelogram that is not a rectangle. Have them identify the center of each figure by finding the intersection of the diagonals. Ask students to place a pin through the center of each figure, and then rotate the figures about the pins to test for rotational symmetry.


194 Chapter 4



1. COMPLETE THE SENTENCE When a point (a, b) is rotated counterclockwise about the origin,

(a, b) → (b, −a) is the result of a rotation of ______.

2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

What are the coordinates of the vertices of the image after a

90° counterclockwise rotation about the origin?


270° clockwise rotation about the origin?

What are the coordinates of the vertices of the image after

turning the fi gure 90° to the left about the origin?


270° counterclockwise rotation about the origin?


In Exercises 3–6, trace the polygon and point P. Then draw a rotation of the polygon about point P using the given number of degrees. (See Example 1.)

3. 30° 4. 80°

A

B

C

P

D

G F

EP

5. 150° 6. 130°

J

G

F

P

R

Q

P

In Exercises 7–10, graph the polygon and its image after a rotation of the given number of degrees about the origin. (See Example 2.)

7. 90°

x

y

4

42−2−4

A

B

C

8. 180°

D

E

F

x

y

−2

4

9. 180° 10. 270°

M L

KJ

x

y

4

2

42 6

Q T

SRx

y

−2

−6

In Exercises 11–14, graph — XY with endpoints X(−3, 1) and Y(4, −5) and its image after the composition. (See Example 3.)

11. Translation: (x, y) → (x, y + 2)


12. Rotation: 180° about the origin




14. Refl ection: in the line y = xRotation: 180° about the origin


A

B

C

x

y4

2

−4

−2

42−2−4


ANSWERS1. 270°

2. What are the coordinates of the

vertices of the image after a 270°

counterclockwise rotation about the

origin?;

A′(2, −1), B′(4,−2), C′(2, −4);

A′(−2, 1), B′(−4, 2), C′(−2, 4)

3.

4.

5.

6.

B′C′

A′A

B

C

P

D

G F

F′

G′D′

E E′

P

F

G

J′

F′J

G′

P

R

Q′

R′Q

P


ASSIGNMENT

Basic: 1, 2, 3–25 odd, 30, 38, 40, 41

Average: 1, 2, 4–32 even, 35, 38, 40, 41

Advanced: 1, 2, 6, 10, 15, 24–34, 36–41

HOMEWORK CHECK

Basic: 2, 7, 13, 17, 25

Average: 2, 10, 16, 26, 32

Advanced: 10, 15, 29, 34, 36

7.

8.

x

y4

2

1

−3

−2

321−2−4

B′

A′

C′B

CA

x

y

1

−2

3 41−4

D′

E′

F′

F

E

D

9.


x

y

4

5

2

3

1

−3

−5

4 5 6 7321−2−1−3−4−5−6−7

L′ M′

J′

K′

K

LM

J










Section 4.3 195


In Exercises 15 and 16, graph △LMN with vertices L(1, 6), M(−2, 4), and N(3, 2) and its image after the composition. (See Example 3.)



16. Refl ection: in the x-axis


In Exercises 17–20, determine whether the fi gure has rotational symmetry. If so, describe any rotations that map the fi gure onto itself. (See Example 4.)

17. 18.

19. 20.

REPEATED REASONING In Exercises 21–24, select the angles of rotational symmetry for the regular polygon. Select all that apply.

○A 30° ○B 45° ○C 60° ○D 72°

○E 90° ○F 120° ○G 144° ○H 180°

21. 22.

23. 24.

ERROR ANALYSIS In Exercises 25 and 26, the endpoints of — CD are C(−1, 1) and D(2, 3). Describe and correct the error in fi nding the coordinates of the vertices of the image after a rotation of 270° about the origin.

25. C (−1, 1) → C ′ (−1, −1) D (2, 3) → D ′ (2, −3)✗

26. C (−1, 1) → C ′ (1, −1) D (2, 3) → D ′ (3, 2)✗

27. CONSTRUCTION Follow these steps to construct a

rotation of △ABC by angle D around a point O. Use

a compass and straightedge.

O

DC

BAA′

Step 1 Draw △ABC, ∠D, and O, the center

of rotation.

Step 2 Draw — OA . Use the construction for copying

an angle to copy ∠D at O, as shown. Then

use distance OA and center O to fi nd A′.

Step 3 Repeat Step 2 to fi nd points B′ and C′. Draw

△A′B′C′.

28. REASONING You enter the revolving door at a hotel.

a. You rotate the door 180°. What does this mean in the

context of the situation?

Explain.

b. You rotate the door 360°. What does this mean in the

context of the situation?

Explain.

29. MATHEMATICAL CONNECTIONS Use the graph of

y = 2x − 3.

a. Rotate the line 90°, 180°, 270°, and 360° about the

origin. Write the equation

of the line for each image.

Describe the relationship

between the equation of the

preimage and the equation

of each image.

b. Do you think that the relationships you

described in part (a) are true for any line?


30. MAKING AN ARGUMENT Your friend claims that

rotating a fi gure by 180° is the same as refl ecting a

fi gure in the y-axis and then refl ecting it in the x-axis.

Is your friend correct? Explain your reasoning.

x

y

−2

2−2


ANSWERS15.

16.

17. yes; Rotations of 90° and 180° about

the center map the fi gure onto itself.



19. yes; Rotations of 45°, 90°, 135°, and

180° about the center map the fi gure

onto itself.

20. yes; A 180° rotation about the center

maps the rectangle onto itself.

21. F

22. E, H

23. D, G

24. C, F, H

25. The rule for a 270° rotation,

(x, y) → (y, −x), should have

been used instead of the rule for

a refl ection in the x-axis;

C(−1, 1) → C′(1, 1),

D(2, 3) → D′(3, −2)

26. The rule for a 270° rotation,

(x, y) → (y, – x), should have been used

instead of the rule for a refl ection in the

line y = x; C(−1, 1) → C′(1, 1),

D(2, 3) → D′(3, −2)

27.

x

y

3

4

6

1

2

−2

2 31−2−4−8−9

M′

N′L′

L″

N″

M″

M

L

N

x

y

3

4

6

1

2

−2

−4

−6

2 31−2−1−4

M′

N′

L′

L″N″

M″

M

L

N

D

D

A

A′

B′

C′

B

C

28. a. If you were outside, you are now inside,

or vice versa, because you have made

half of a rotation.

b. You are back where you started because

you have made a full rotation.


30. yes; Refl ection in the y-axis and then in the

x-axis yields (x, y) → (−x, y) → (−x, −y).

A 180° rotation yields the same result:

(x, y) → (−x, −y).


196 Chapter 4


31. DRAWING CONCLUSIONS A fi gure only has point

symmetry. How many times can you rotate the fi gure

before it is back where it started?

32. ANALYZING RELATIONSHIPS Is it possible for a

fi gure to have 90° rotational symmetry but not 180° rotational symmetry? Explain your reasoning.

33. ANALYZING RELATIONSHIPS Is it possible for a

fi gure to have 180° rotational symmetry but not 90° rotational symmetry? Explain your reasoning.

34. THOUGHT PROVOKING Can rotations of 90°, 180°, 270°, and 360° be written as the composition of

two refl ections? Justify your answer.

35. USING AN EQUATION Inside a kaleidoscope, two

mirrors are placed next to each other to form a V. The

angle between the mirrors determines the number of

lines of symmetry in the

image. Use the formula

n(m∠1) = 180° to fi nd the

measure of ∠1, the angle

between the mirrors, for the

number n of lines of symmetry.

a. b.

36. REASONING Use the coordinate rules for

counterclockwise rotations about the origin to write

coordinate rules for clockwise rotations of 90°, 180°, or 270° about the origin.

37. USING STRUCTURE △XYZ has vertices X(2, 5),

Y(3, 1), and Z(0, 2). Rotate △XYZ 90° about the point

P(−2, −1).

38. HOW DO YOU SEE IT? You are fi nishing the puzzle.

The remaining two pieces both have rotational

symmetry.

1 2

a. Describe the rotational symmetry of Piece 1 and

of Piece 2.

b. You pick up Piece 1. How many different ways

can it fi t in the puzzle?

c. Before putting Piece 1 into the puzzle, you

connect it to Piece 2. Now how many ways can

it fi t in the puzzle? Explain.

39. USING STRUCTURE A polar coordinate system locates

a point in a plane by its distance from the origin O

and by the measure of an angle with its vertex at the

origin. For example, the point A(2, 30°) is 2 units

from the origin and m∠XOA = 30°. What are the

polar coordinates of the image of point A after a 90° rotation? a 180° rotation? a 270° rotation? Explain.

90°60°

30°

X 0°3

330°

300°270°

240°

210°

180°

150°

120°

1 2

A

O

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyThe fi gures are congruent. Name the corresponding angles and the corresponding sides. (Skills Review Handbook)

40. PQ

RS

T

VW

X

YZ

41. A B

CDJ K

LM


1mirror

black glass


Mini-Assessment

1. Draw a 135° rotation of △ ABC about point P.

A

B

C

P

See Additional Answers.

2. — RS has endpoints R(−2, 1) and S(1, 2). Graph — RS and its image after the composition.

Rotation: 90° about the origin Reflection: in the x-axis

x

y

2

2−4

R′

S′

R″

S″

RS

3. Does the figure have rotational symmetry? If so, describe any rotations that map the figure onto itself.

yes; 90°, 180°

ANSWERS31. twice

32. no; Because the fi gure has 90°

rotational symmetry, the image will

still be symmetrical to the preimage

after two 90° rotations, which is the

equivalent of a 180° rotation.

33. yes; Sample answer: A rectangle

(that is not a square) is one example

of a fi gure that has 180° rotational

symmetry, but not 90° rotational

symmetry.


35. a. 15°, n = 12

b. 30°, n = 6

36. (x, y) → (y, −x); (x, y) → (−x, −y);

(x, y) → (−y, x)

















Chapter 4 197

197197

4.1–4.3 What Did You Learn?

Core VocabularyCore Vocabularyvector, p. 174initial point, p. 174terminal point, p. 174horizontal component, p. 174vertical component, p. 174component form, p. 174transformation, p. 174image, p. 174

preimage, p. 174translation, p. 174rigid motion, p. 176composition of transformations,

p. 176refl ection, p. 182line of refl ection, p. 182glide refl ection, p. 184

line symmetry, p. 185line of symmetry, p. 185rotation, p. 190center of rotation, p. 190angle of rotation, p. 190rotational symmetry, p. 193center of symmetry, p. 193

Core ConceptsCore ConceptsSection 4.1Vectors, p. 174Translations, p. 174

Postulate 4.1 Translation Postulate, p. 176Theorem 4.1 Composition Theorem, p. 176

Section 4.2Refl ections, p. 182Coordinate Rules for Refl ections, p. 183

Postulate 4.2 Refl ection Postulate, p. 184Line Symmetry, p. 185

Section 4.3Rotations, p. 190Coordinate Rules for Rotations

about the Origin, p. 191

Postulate 4.3 Rotation Postulate, p. 192Rotational Symmetry, p. 193

Mathematical PracticesMathematical Practices1. How could you determine whether your results make sense in Exercise 26 on page 179?

2. State the meaning of the numbers and symbols you chose in Exercise 28 on page 187.

3. Describe the steps you would take to arrive at the answer to Exercise 29 part (a) on page 195.

Ever feel frustrated or overwhelmed by math? You’re not alone. Just take a deep breath and assess the situation. Try to fi nd a productive study environment, review your notes and examples in the textbook, and ask your teacher or peers for help.

Study Skills

Keeping a Positive Attitude

HSCC_GEOM_PE_04.MC.indd 197 2/24/14 10:42 AM

ANSWERS1. Recreate the chess board on a

coordinate plane and substitute the

coordinates into your rule to verify

both the composition and the single

translation yield the same result.

2. x + 3 means that the fi gure will slide

3 units to the right, and y + 3 means

the fi gure will slide 3 units up.

3. Find two points on the line

y = 2x − 3, their images after the

rotation, and use the images to fi nd

the equation of the new line.

HSCC_GEOM_TE_04MC.indd 197HSCC_GEOM_TE_04MC.indd 197 6/5/14 1:52 PM6/5/14 1:52 PM

198 Chapter 4


4.1– 4.3 Quiz

Graph quadrilateral ABCD with vertices A(−4, 1), B(−3, 3), C(0, 1), and D(−2, 0) and its image after the translation. (Section 4.1)

1. (x, y) → (x + 4, y − 2) 2. (x, y) → (x − 1, y − 5) 3. (x, y) → (x + 3, y + 6)

Graph the polygon with the given vertices and its image after a refl ection in the given line. (Section 4.2)

4. A(−5, 6), B(−7, 8), C(−3, 11); x-axis 5. D(−5, −1), E(−2, 1), F(−1, −3); y = x

6. J(−1, 4), K(2, 5), L(5, 2), M(4, −1); x = 3 7. P(2, −4), Q(6, −1), R(9, −4), S(6, −6); y = −2

Graph △ABC with vertices A(2, −1), B(5, 2), and C(8, −2) and its image after the glide refl ection. (Section 4.2)

8. Translation: (x, y) → (x, y + 6) 9. Translation: (x, y) → (x − 9, y)

Refl ection: in the y-axis Refl ection: in the line y = 1

Determine the number of lines of symmetry for the fi gure. (Section 4.2)

10. 11. 12. 13.

Graph the polygon and its image after a rotation of the given number of degrees about the origin. (Section 4.3)

14. 90°

x

y4

2

−4

−2

42−2−4A C

B 15. 270°

x

y

−4

−2

42−2−4G

D

E

F

16. 180°

x

y4

2

−4

42

J

H

K

I

Graph △LMN with vertices L(−3, −2), M(−1, 1), and N(2, −3) and its image after the composition. (Sections 4.1–4.3)

17. Translation: (x, y) → (x − 4, y + 3)




19. The fi gure shows a game in which the object is to create solid rows

using the pieces given. Using only translations and rotations, describe the

transformations for each piece at the top that will form two solid rows at

the bottom. (Section 4.1 and Section 4.3)

x

y

HSCC_GEOM_PE_04.MC.indd 198 2/24/14 10:42 AM

ANSWERS1.

2.

3.

4.

5.

6.


10. 6

11. 0

12. 2

13. 1

x

y3

2

−2

2 41−3 −1−4

B′

A′D′

C′

A

B

C

D

x

y3

2

−2

−3

−5

−3 −1−4−5

B′

A′D′

C′

A

B

C

D

x

3

4

5

6

9

2

−3 −1 321−4

B′

A′D′

C′

A

B

C

D

x

y

6

8

10

4

2

−4

−6

−8

−10

−6 −2−4−8

12

A

A′

B

B′

C

C′

x−2

−2

y

D

D′

E′F′

E

Fy = x

x

5

2

6

y

J′

K′

L′

M′

L

M

J

K

x = 3

14.

15.

x

4

2

42−2−4

y B

CA′

B′

C′

A

x

4

2

42−2

y

−2

D

D′

E′

F′G′

E

F

G

16.


x

4

2

2−2

y

−4

H

J′

I′H′

K′

I

J

K

HSCC_GEOM_TE_04MC.indd 198HSCC_GEOM_TE_04MC.indd 198 6/5/14 1:52 PM6/5/14 1:52 PM











Section 4.4 T-198

Laurie’s Notes

Overview of Section 4.4Introduction• In grade 8, the concept of congruency was introduced. Students should understand that a

two-dimensional figure is congruent to another when the second can be obtained from the first by a sequence of rotations, reflections, and translations.

• In this chapter, students have studied three rigid transformations, all congruence preserving in terms of length and angle measure. Students learned that a composition of rigid motions will also be a rigid motion. Two specific compositions are explored in this lesson: successive reflections in parallel lines and successive reflections in intersecting lines. The results are stated as theorems.

Resources• MP5 Use Appropriate Tools Strategically: To enable students to see the results of two

or more rigid transformations, use dynamic geometry software, tracing paper, transparencies, and reflecting devices. At home, students can use parchment paper or waxed paper.

Teaching Strategy• The coordinate plane provides a calibrated environment, which aids students’ visual skills

when performing a rigid transformation on a figure. When using dynamic geometry software, transformations can be done on a coordinate grid or not.

• You can model both environments without software! Use an overhead projector and two layers of transparencies, one that has a coordinate grid on it and one that is clear.

• Draw the figures from Example 1 or 2 on the clear transparency and overlay on the coordinate grid. Have students identify the congruent figures and describe the transformations. Use tick marks to indicate lengths that are congruent. Slide the coordinate grid out from under the top transparency.

“Are the figures congruent?” yes

Extensions• Natural extensions to this lesson would be to perform more than two reflections (in parallel

or intersecting lines) and look for patterns.

Pacing Suggestion• In the explorations, the students explore the two theorems presented in the lesson. The

discussion and examples in the formal lesson related to the theorems will require less time.


Laurie’s Notes

T-199 Chapter 4

ExplorationMotivate• Fold a piece of paper in half and in half again so that

the fold lines are parallel, as shown. Use a handheld paper punch to punch one hole through all layers. Have students describe what the paper will look like when opened.

• Fold a piece of paper in half and in half again so that the fold lines are perpendicular, as shown. Use a handheld paper punch to punch one hole through all layers. Have students describe what the paper will look like when opened.

• Discuss the location of the lines of reflection on the opened paper. Explain to students that today they will be investigating reflections in lines.

Exploration Note• Each exploration investigates a theorem presented in the formal lesson. It is possible to

simply pose the conditions to students and let them explore.

Exploration 1• The correspondence of sides is easier to see when a scalene triangle is used. Note that the

coordinate grid is not necessary in this exploration.• If you want students to discover the relationship, then I think it is best when students are not

looking at a graphic of the finished construction. Just give brief instructions.• Say, “Construct a scalene triangle on the left side of the screen. Draw two parallel lines

that you can reflect your triangle in. Reflect your triangle in the first line followed by the second line. I’m curious about the relationship between the original triangle and the final triangle.”

• Depending on how far apart the parallel lines are, or how far the triangle is from the first line, the resulting construction can be more difficult for students to read and interpret. Fortunately, with the dynamic geometry software the position of the parallel lines can be changed easily.

• It is actually okay when the reflected triangles intersect either of the parallel lines. The conjecture you want students to make will not change.

Probing Question: If students need a little assistance, say, “Hide the middle triangle, the image of the first reflection. What do you notice?” At this point, students should recognize that the final image is a translation of the original triangle.

“What is the vector for this translation?” The vector is two times the distance between the parallel lines. If students do not observe the connection of the vector to the distance between the parallel lines, suggest that they click and drag on one of the parallel lines.

Exploration 2• I prefer to have students explore without the benefit of a diagram.

• To introduce the exploration, ask, “What if the lines in Exploration 1 were not parallel? Would there be a relationship between the original triangle and the final image?”

• Ask probing questions to assist students’ discovery of the relationship being explored.

Communicate Your AnswerCommunicate Your Answer• MP5: If students are having difficulty with Question 4, suggest that they draw a sketch.

Connecting to Next Step• The explorations students have explored will be presented in the formal lesson as theorems.


HSG-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, … or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HSG-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Parallelfold lines

Perpendicularfold lines












Section 4.4 199

Section 4.4 Congruence and Transformations 199

Congruence and Transformations4.4

Refl ections in Parallel Lines

Work with a partner. Use dynamic geometry software to draw any scalene triangle

and label it △ABC.

a. Draw any line �� DE . Refl ect △ABC

in �� DE to form △A′B′C′.

b. Draw a line parallel to �� DE . Refl ect

△A′B′C′ in the new line to form

△A″B″C″.

c. Draw the line through point A that

is perpendicular to �� DE . What do

you notice?

d. Find the distance between points A

and A″. Find the distance between

the two parallel lines. What do

you notice?

e. Hide △A′B′C′. Is there a single transformation that maps △ABC to △A″B″C″?

Explain.

f. Make conjectures based on your answers in parts (c)–(e). Test your conjectures

by changing △ABC and the parallel lines.

Refl ections in Intersecting Lines

Work with a partner. Use dynamic geometry software to draw any scalene triangle

and label it △ABC.

a. Draw any line �� DE . Refl ect △ABC

in �� DE to form △A′B′C′.

b. Draw any line �� DF so that angle

EDF is less than or equal to 90°. Refl ect △A′B′C′ in �� DF to form

△A″B″C″.c. Find the measure of ∠EDF.

Rotate △ABC counterclockwise

about point D using an angle

twice the measure of ∠EDF.

d. Make a conjecture about a fi gure

refl ected in two intersecting lines.

Test your conjecture by changing

△ABC and the lines.

Communicate Your AnswerCommunicate Your Answer 3. What conjectures can you make about a fi gure refl ected in two lines?

4. Point Q is refl ected in two parallel lines, �� GH and �� JK , to form Q′ and Q″. The distance from �� GH to �� JK is 3.2 inches. What is the distance QQ″?

Essential QuestionEssential Question What conjectures can you make about a fi gure

refl ected in two lines?COMMON CORE

Learning StandardHSG-CO.A.5HSG-CO.B.6

CONSTRUCTING VIABLE ARGUMENTS

To be profi cient in math, you need to make conjectures and justify your conclusions.

A

C

E

D

F

B

A′

A″

B″C″

C′

B′

Sample

A

C

E

D

F

B

A′ A″

B″C″

C′

B′

Sample


ANSWERS1. a. Check student’s work.

b. Check student’s work.

c. Sample answer:

The line passes through A′ and A″. d. The distance between A and A″

is twice the distance between the

parallel lines.

e. yes; △A″B″C″ is a translation of

△ABC.

f. If two lines are parallel, and a

preimage is refl ected in the fi rst

line and then in the second, the

fi nal image is a translation of the

preimage. The distance between

each point in the preimage and

its corresponding point in the

fi nal image is twice the distance

between the parallel lines.

2. a. Check student’s work.

b. Check student’s work.

c. Sample answer: 50° d. The fi nal image after the

refl ections is the same as a

rotation about point D using an

angle that is twice the measure

of the angle of intersection.

3. The image of a fi gure refl ected

in two lines is congruent to the

preimage. The image of a fi gure

refl ected in two parallel lines is a

translation of the preimage. The

image of a fi gure refl ected in two

lines that intersect in point D is a

rotation in point D of the preimage.

4. 6.4 in.

AD

E

F

A′

B′

C′

C ″

B

CA″

B″


200 Chapter 4


4.4 Lesson What You Will LearnWhat You Will Learn Identify congruent fi gures.

Describe congruence transformations.

Use theorems about congruence transformations.

Identifying Congruent FiguresTwo geometric fi gures are congruent fi gures if and only if there is a rigid motion or

a composition of rigid motions that maps one of the fi gures onto the other. Congruent

fi gures have the same size and shape.

Congruent

same size and shape

Not congruent

different sizes or shapes

You can identify congruent fi gures in the coordinate plane by identifying the rigid

motion or composition of rigid motions that maps one of the fi gures onto the other.

Recall from Postulates 4.1–4.3 and Theorem 4.1 that translations, refl ections,

rotations, and compositions of these transformations are rigid motions.

Identifying Congruent Figures

Identify any congruent fi gures in the

coordinate plane. Explain.

SOLUTIONSquare NPQR is a translation of square ABCD

2 units left and 6 units down. So, square ABCD

and square NPQR are congruent.

△KLM is a refl ection of △EFG in the x-axis.

So, △EFG and △KLM are congruent.

△STU is a 180° rotation of △HIJ.

So, △HIJ and △STU are congruent.


1. Identify any congruent fi gures in the

coordinate plane. Explain.

congruent fi gures, p. 200congruence transformation,

p. 201


x

y5

−5

5

I H

J

G E

F

C B

AD

Q P

NR

S T

U

M

L

K

x

y

4

42−2−4

I H

J G

E

FC

B

A

D

Q P

NR

S

T

UM

L

K


Extra Example 1Identify any congruent figures in the coordinate plane. Explain.

AP R

QS U

T

C

B

x

y

4

6

2

4 62−2−4

△ ABC is congruent to △ PQR, because△ PQR is a translation of △ ABC 5 units right and 1 unit up.

Teacher ActionsTeacher ActionsLaurie’s Notes What does it mean for two figures to be congruent?” Answers will vary, but it is likely that students will say things such as “same shape and same size,” or “all the corresponding sides are congruent and all corresponding angles are congruent.” These responses reflect an understanding of congruence from middle school.

• Define congruent figures in terms of a rigid motion. Note the “if and only if” language of the definition.

• FYI: △ STU is also a reflection of △ JIH in the line y = x.• MP3 Construct Viable Arguments and Critique the Reasoning of Others: Have

students Turn and Talk to answer the Monitoring Progress question. What evidence are partners offering? They should be stating a rigid motion that mapped one figure to another.


KinestheticStudents may not fully understand that only rigid transformations produce congruent figures. Have students take a sheet of paper and transform it in various ways. Students should not only slide, spin, and flip the paper, but they should also cut, fold, or crumple the paper into a ball. After each transformation, ask students to explain how they decide whether the transformed paper is congruent to its original shape. Encourage students to use the terms sides, vertices, length, distance, or angles in their responses.

MONITORING PROGRESS ANSWER 1. △DEF ≅ △ABC, △KLM ≅ △STU,

▭GHIJ ≅ ▭NPQR; △DEF is a

90° rotation of △ABC. △KLM is a

refl ection of △STU in the y-axis.

▭GHIJ is a translation 6 units up

of ▭NPQR.


Section 4.4 201


Congruence TransformationsAnother name for a rigid motion or a combination of rigid motions is a congruence transformation because the preimage and image are congruent. The terms “rigid

motion” and “congruence transformation” are interchangeable.

Describing a Congruence Transformation

Describe a congruence transformation

that maps ▱ABCD to ▱EFGH.

SOLUTIONThe two vertical sides of ▱ABCD rise from left

to right, and the two vertical sides of ▱EFGH fall

from left to right. If you refl ect ▱ABCD in the

y-axis, as shown, then the image, ▱A′B′C′D′, will have the same orientation as ▱EFGH.

Then you can map ▱A′B′C′D′ to ▱EFGH using a translation of 4 units down.

So, a congruence transformation that maps ▱ABCD to ▱EFGH is a refl ection

in the y-axis followed by a translation of 4 units down.


2. In Example 2, describe another congruence transformation that

maps ▱ABCD to ▱EFGH.

3. Describe a congruence transformation that maps △JKL to △MNP.

x

y4

−4

−2

42−2−4

L J

K

P M

N

READINGYou can read the notation ▱ABCD as “parallelogram A, B, C, D.”

x

y4

2

−2

42

C

BA

D

F E

HG

x

y4

−2

42

C

BA

DD′

A′B′

C′

F E

HG


Extra Example 2Describe a congruence transformation that maps quadrilateral ABCD to quadrilateral PQRS.

x

y2

−6

−2

P Q

R

SDA

BC

90° rotation about the origin


Group ActivitySome students may struggle with the fact that the terms congruence transformation and rigid motion are interchangeable. Form small groups of English learners and English speakers. Have each group find five sets of interchangeable terms (mathematical or otherwise). Ask groups to share their findings. As needed, discuss the difference between interchangeable terms and a pair of terms where one describes a subset of the other (such as square and quadrilateral).

Teacher ActionsTeacher ActionsLaurie’s Notes• Define congruence transformation. The term isometry could also be used.• Pose Example 2 as given. Check with a Thumbs Up that students understand the problem

statement. Give time for partners to work together.• Popsicle Sticks: Solicit a solution from the class. “Did everyone have that solution?” There are

different compositions of rigid motions that will work.• Extension: Have students explore whether a translation followed by a reflection is

commutative, meaning does the order in which the rigid motions are performed matter?


2. Sample answer: refl ection in the

x-axis followed by a translation

5 units left


x-axis followed by a translation

5 units right


202 Chapter 4


Using Theorems about Congruence TransformationsCompositions of two refl ections result in either a translation or a rotation. A

composition of two refl ections in parallel lines results in a translation, as described in

the following theorem.

Using the Refl ections in Parallel Lines Theorem

In the diagram, a refl ection in line k

maps — GH to

— G′H′ . A refl ection in line m

maps — G′H′ to

— G″H″ . Also, HB = 9

and DH″ = 4.

a. Name any segments congruent to

each segment: — GH , — HB , and

— GA .

b. Does AC = BD? Explain.

c. What is the length of — GG″ ?

SOLUTION

a. — GH ≅

— G′H′ , and — GH ≅

— G″H″ . — HB ≅ — H′B . — GA ≅

— G′A .

b. Yes, AC = BD because — GG″ and

— HH″ are perpendicular to both k and m. So, — BD

and — AC are opposite sides of a rectangle.

c. By the properties of refl ections, H′B = 9 and H′D = 4. The Refl ections in Parallel

Lines Theorem implies that GG″ = HH″ = 2 ⋅ BD, so the length of — GG″ is

2(9 + 4) = 26 units.


Use the fi gure. The distance between line k and line m is 1.6 centimeters.

4. The preimage is refl ected in line k, then in

line m. Describe a single transformation that

maps the blue fi gure to the green fi gure.

5. What is the relationship between — PP′ and

line k? Explain.

6. What is the distance between P and P ″?

TheoremTheoremTheorem 4.2 Refl ections in Parallel Lines Theorem

If lines k and m are parallel, then a refl ection in

line k followed by a refl ection in line m is the

same as a translation.

If A″ is the image of A, then

1. — AA″ is perpendicular to k and m, and

2. AA″ = 2d, where d is the distance

between k and m.

Proof Ex. 31, p. 206

kB

A

d

B′ B″

A″A′

m

k

H

B

A

D

CG

H′ H″

G″G′m

k

P

m

P″P′


Extra Example 3In the diagram, a reflection in line a maps — PQ to — P′Q′ . A reflection in line b maps — P′Q′ to — P″Q″ . Also, PJ = 3 and LP″ = 8.

aP

Q

P′ P″

Q″Q′

b

K

LJ

M

a. Name any segments congruent to each segment: — PQ , — PJ , and — QK .

— PQ ≅ — P′Q′ ≅ — P″Q″ ; — PJ ≅ — P′J ; — QK ≅ — Q′K b. Does JK = LM? Explain. Yes, JKML is

a rectangle, and opposite sides of a rectangle are congruent.

c. What is the length of — PP″ ? 22 units

Teacher ActionsTeacher ActionsLaurie’s Notes• The Reflections in Parallel Lines Theorem (Thm. 4.2) was explored in Exploration 1. Whether

the exploration was done or not, use dynamic geometry software to demonstrate the result of composing two reflections in parallel lines. Click and drag the parallel lines, changing the distance d between them and hence the distance 2d between the preimage and image.

• MP5: Students need to have the opportunity to explore the Reflections in Parallel Lines Theorem (Thm. 4.2) with technology, reflectors, or compass and straightedge. Students should develop the habit of mind of verifying statements using appropriate tools.

• Alternate Approach: For Example 3, display the diagram. Give partners a whiteboard and ask them to record everything they know to be true from the diagram. They should be prepared to justify their statements.


4. translation 3.2 cm right

5. They are perpendicular by Refl ections

in Parallel Lines (Thm. 4.2).

6. 3.2 cm


Section 4.4 203


Using the Refl ections in Intersecting Lines Theorem

In the diagram, the fi gure is refl ected in line k. The image is then refl ected in line m.

Describe a single transformation that maps F to F ″.

FP

m

kF″ F′70°

SOLUTION

By the Refl ections in Intersecting Lines Theorem, a refl ection in line k followed by a

refl ection in line m is the same as a rotation about point P. The measure of the acute

angle formed between lines k and m is 70°. So, by the Refl ections in Intersecting

Lines Theorem, the angle of rotation is 2(70°) = 140°. A single transformation that

maps F to F ″ is a 140° rotation about point P.

You can check that this is correct by tracing lines k and m and point F, then

rotating the point 140°.


7. In the diagram, the preimage is refl ected in

line k, then in line m. Describe a single

transformation that maps the blue fi gure

onto the green fi gure.

8. A rotation of 76° maps C to C′. To map C

to C′ using two refl ections, what is the

measure of the angle formed by the

intersecting lines of refl ection?

TheoremTheoremTheorem 4.3 Refl ections in Intersecting Lines TheoremIf lines k and m intersect at point P, then a

refl ection in line k followed by a refl ection

in line m is the same as a rotation about

point P.

The angle of rotation is 2x°, where x° is

the measure of the acute or right angle

formed by lines k and m.

m∠BPB″ = 2x°Proof Ex. 31, p. 250

k

BAP

B′B″

A″A′

x°2x°

m

P

m

k

80°

A composition of two refl ections in intersecting lines results in a rotation, as described

in the following theorem.


Extra Example 4In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps F to F″.

F

P m

k

F″

F′

58°

A rotation of 116° about point P

Teacher ActionsTeacher ActionsLaurie’s Notes• The Reflections in Intersecting Lines Theorem (Thm. 4.3) is more

difficult to see for most students than the previous theorem. Rotations in general are more difficult to visualize than translations.

• Use software to demonstrate the result of composing two reflections in intersecting lines. Click and drag either of the intersecting lines, changing x°, the acute or right angle formed by the intersecting lines, and hence, the angle of rotation 2x° between the preimage and image.

• Note: A reflection in the x-axis followed by a reflection in the y-axis is equivalent to a rotation of 180°. This special case is one that should be familiar to students.

• Teaching Tip: Using a figure that is not a polygon, like Example 4, can help students view the orientation that changes when a figure is reflected.

ClosureClosure• Writing Prompt: A congruence transformation is … a rigid

motion where the preimage and image are congruent.• Writing Prompt: The composition of two reflections is equivalent

to … a translation when the lines are parallel and is equivalent to a rotation when the lines are intersecting.


7. 160° rotation about point P

8. 38°


204 Chapter 4



1. COMPLETE THE SENTENCE Two geometric fi gures are _________ if and only if there is a rigid motion

or a composition of rigid motions that moves one of the fi gures onto the other.

2. VOCABULARY Why is the term congruence transformation used to refer to a rigid motion?


In Exercises 3 and 4, identify any congruent fi gures in the coordinate plane. Explain. (See Example 1.)

3.

x

y

4

2

−4

−2

4−2−4

J

KHM N

PL

B

A C T V

U GD

E F

S

Q R

4.

x

y

−4

−2

−3J K

H

M

NP

L

B

A

C

T V

U

G

D

E

F

S Q

R

In Exercises 5 and 6, describe a congruence transformation that maps the blue preimage to the green image. (See Example 2.)

5.

x

y4

2

42−2−4−6

A

CB

G F

E

6.

x

y

4

−4

−2

4 62−2−4−6S

PQ

R

W X

YZ

In Exercises 7–10, determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning.

7. Q(2, 4), R(5, 4), S(4, 1) and T(6, 4), U(9, 4), V(8, 1)

8. W(−3, 1), X(2, 1), Y(4, −4), Z(−5, −4) and

C(−1, −3), D(−1, 2), E(4, 4), F(4, −5)

9. J(1, 1), K(3, 2), L(4, 1) and M(6, 1), N(5, 2), P(2, 1)

10. A(0, 0), B(1, 2), C(4, 2), D(3, 0) and

E(0, −5), F(−1, −3), G(−4, −3), H(−3, −5)

In Exercises 11–14, k � m, △ABC is refl ected in line k, and △A′B′C′ is refl ected in line m. (See Example 3.)

11. A translation maps

△ABC onto which

triangle?

12. Which lines are

perpendicular to — AA″ ?

13. If the distance between

k and m is 2.6 inches,

what is the length of — CC ″ ?

14. Is the distance from B′ to m the same as the distance

from B ″ to m? Explain.


k

B

C

A

B′ B″

A″

C″

A′

C′

m


ANSWERS 1. congruent

2. The preimage and image are

congruent in a rigid transformation.

3. △HJK ≅ △QRS, ▭DEFG ≅

▭LMNP; △HJK is a 90° rotation of

△QRS. ▭DEFG is translation 7 units

right and 3 units down of ▭LMNP.

4. △MNP ≅ △TUV, △EFG ≅ △QRS, ▭HJKL ≅ ▭ABCD; △MNP is a

90° rotation of △TUV. △EFG is a

180° rotation of △QRS. ▭HJKL is a

translation 4 units down and 7 units

right of ▭ABCD.

5. Sample answer: 180° rotation about

the origin followed by a translation

5 units left and 1 unit down


the origin

7. yes; △TUV is a translation 4 units

right of △QRS. So, △TUV ≅ △QRS.

8. yes; CDEF is a 90° rotation of

WXYZ. So, CDEF ≅ WXYZ.

9. no; M and N are translated 2 units

right of their corresponding vertices,

L and K, but P is translated only 1 unit

right of its corresponding vertex, J.

So, this is not a rigid motion.

10. yes; A congruence transformation

that maps ▱ABCD to ▱GHEF is a

translation 5 units down, followed

by a refl ection in the y-axis. So,

▱ABCD ≅ ▱GHEF.

11. A″B″C″ 12. line k and line m

13. 5.2 in.

14. yes; Because △A″B″C″ is a refl ection

of △A′B′C′ in line m, each vertex in

the image is the same distance from

the line of refl ection as its preimage.


ASSIGNMENT

Basic: 1, 2, 3–21 odd, 30, 33, 37–43

Average: 1, 2, 6–30 even, 33, 37–43

Advanced: 1, 2, 6, 10–18, 21, 23–34, 37–43

HOMEWORK CHECK

Basic: 5, 7, 17, 19, 30

Average: 6, 16, 18, 20, 30

Advanced: 6, 17, 29, 30, 34










Section 4.4 205


In Exercises 15 and 16, fi nd the angle of rotation that maps A onto A″. (See Example 4.)

15. m

k55°

A

A″A′

16.

m

k15°

AA″

A′


describing the congruence transformation.

x

y

2

−2

42−2−4

A C

B

B″C″

A″

△ABC is mapped to △A″B ″C ″ by a translation 3 units down and a refl ection in the y-axis.

✗


using the Refl ections in Intersecting Lines Theorem

(Theorem 4.3).

P

72°

A 72° rotation about point P maps the blue image to the green image.

✗

In Exercises 19–22, fi nd the measure of the acute or right angle formed by intersecting lines so that C can be mapped to C′ using two refl ections.

19. A rotation of 84° maps C to C′.

20. A rotation of 24° maps C to C′.

21. The rotation (x, y) → (−x, −y) maps C to C′.

22. The rotation (x, y) → (y, −x) maps C to C′.

23. REASONING Use the Refl ections in Parallel Lines

Theorem (Theorem 4.2) to explain how you can make

a glide refl ection using three refl ections. How are the

lines of refl ection related?

24. DRAWING CONCLUSIONS The pattern shown is

called a tessellation.

a. What transformations did the artist use when

creating this tessellation?

b. Are the individual fi gures in the tessellation

congruent? Explain your reasoning.

CRITICAL THINKING In Exercises 25–28, tell whether the statement is always, sometimes, or never true. Explain your reasoning.

25. A congruence transformation changes the size of

a fi gure.

26. If two fi gures are congruent, then there is a rigid

motion or a composition of rigid motions that maps

one fi gure onto the other.

27. The composition of two refl ections results in the

same image as a rotation.

28. A translation results in the same image as the

composition of two refl ections.

29. REASONING During a presentation, a marketing

representative uses a projector so everyone in the

auditorium can view the advertisement. Is this

projection a congruence transformation? Explain

your reasoning.


ANSWERS15. 110°16. 30°17. A translation 5 units right and a

refl ection in the x-axis should have

been used; △ABC is mapped to

△A′B′C′ by a translation 5 units

right, followed by a refl ection in the

x-axis.

18. If x° is the measure of the acute angle

formed by the intersecting lines,

an angle of 2x° should be used to

describe the angle of rotation; A 144° rotation about point P maps the blue

image to the green image.

19. 42° 20. 12° 21. 90°

22. 45° 23. Refl ect the fi gure in two parallel

lines instead of translating the

fi gure; The third line of refl ection is

perpendicular to the parallel lines.

24. a. rotations and translations

b. yes; All of the fi gures could be

created using one or more rigid

transformations of an original

shape.

25. never; Congruence transformations

are rigid motions.

26. always; Every fi gure can be mapped

onto a congruent fi gure using

transformations.

27. sometimes; Refl ecting in y = x then

y = x is not a rotation. Refl ecting in

the y-axis then x-axis is a rotation

of 180°. 28. sometimes; It would depend on the

translations.

29. no; The image on the screen is larger.


206 Chapter 4


30. HOW DO YOU SEE IT? What type of congruence

transformation can be used to verify each statement

about the stained glass window?

1 4

56

8

23

7

a. Triangle 5 is congruent to Triangle 8.

b. Triangle 1 is congruent to Triangle 4.

c. Triangle 2 is congruent to Triangle 7.

d. Pentagon 3 is congruent to Pentagon 6.

31. PROVING A THEOREM Prove the Refl ections in

Parallel Lines Theorem (Theorem 4.2).

K

J

d

K′ K″

J″J′

m

Given A refl ection in lineℓmaps — JK to

— J′K′ , a refl ection in line m maps

— J′K′ to — J ″K″ ,

andℓ � m.

Prove a. — KK ″ is perpendicular toℓand m.

b. KK ″ = 2d, where d is the distance

betweenℓand m.

32. THOUGHT PROVOKING A tessellation is the covering

of a plane with congruent fi gures so that there are no

gaps or overlaps (see Exercise 24). Draw a tessellation

that involves two or more types of transformations.

Describe the transformations that are used to create

the tessellation.

33. MAKING AN ARGUMENT — PQ , with endpoints

P(1, 3) and Q(3, 2), is refl ected in the y-axis.

The image — P′Q′ is then refl ected in the x-axis to

produce the image — P ″Q ″ . One classmate says that

— PQ is mapped to — P ″Q ″ by the translation

(x, y) → (x − 4, y − 5). Another classmate says that — PQ is mapped to

— P ″Q ″ by a (2 ⋅ 90)°, or 180°, rotation

about the origin. Which classmate is correct? Explain

your reasoning.

34. CRITICAL THINKING Does the order of refl ections

for a composition of two refl ections in parallel lines

matter? For example, is refl ecting △XYZ in lineℓand

then its image in line m the same as refl ecting △XYZ

in line m and then its image in lineℓ?

Y

X Z

m

CONSTRUCTION In Exercises 35 and 36, copy the fi gure. Then use a compass and straightedge to construct two lines of refl ection that produce a composition of refl ections resulting in the same image as the given transformation.

35. Translation: △ABC → △A″B ″C ″

A C

BB″

C″A″

36. Rotation about P: △XYZ → △X ″Y ″Z ″

Y PZ

XX″

Z″

Y″

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the equation. Check your solution. (Skills Review Handbook)

37. 5x + 16 = −3x 38. 12 + 6m = 2m 39. 4b + 8 = 6b − 4

40. 7w − 9 = 13 − 4w 41. 7(2n + 11) = 4n 42. −2(8 − y) = −6y

43. Last year, the track team’s yard sale earned $500. This year, the yard sale earned $625. What is the

percent of increase? (Skills Review Handbook)



Mini-Assessment

1. Identify any congruent figures in the coordinate plane. Explain.

x

y

−4

−2

−2 2−4

A C

BJ

D

G F

E

K

MU

L

S T

W V

X

Q

R

P


2. Describe a congruence transformation that maps △ JKL to △ MNP.

x

y4

−2

2−2−4

L J

K

PM

N


3. △TUV is reflected over line k, and then that image is reflected over line m. The distance between lines k and m is 2.8 cm. What is the distance between T and T″?

k m

V

UT

5.6 cm

4. Lines k and m intersect. Figure A is reflected over line k and then that image is reflected over line m to produce figure A″. A 24° rotation about the intersection of lines m and k maps A to A″. What is the measure of the acute angle formed by the lines? 12°







ANSWERS30–43. See Additional Answers.












Section 4.5 T-206

Laurie’s Notes

Overview of Section 4.5Introduction• Dilations were studied in grade 8 where students were expected to describe the effect of

dilations on two-dimensional figures using coordinates. New in this lesson is that the centers of dilation do not need to be at the origin, and students are constructing dilations with a straightedge and a compass.

• The explorations provide an opportunity for students to explore different scale factors and centers of dilation. Encourage students to measure lengths and angles when they are using the dynamic geometry software.

• The scale factor of a dilation is the ratio of corresponding sides of the image and preimage. This ratio is used to solve real-life applications.

Resources• A flashlight can be used to demonstrate the center of dilation. Darken the room. Use the

flashlight to project the image of a figure onto a wall.

Formative Assessment Tips• Think-Alouds: This technique is used when you want to hear how well partners

comprehend a process involved with solving a problem. It is important to model the process first so that students have a sense of what is expected in Think-Alouds.

• Think-Alouds give students the opportunity to hear the metacognitive processes used by someone who is a proficient problem solver.

• MP1 Make Sense of Problems and Persevere in Solving Them and MP6 Attend to Precision: Hearing someone else describe a process using mathematical language will improve all students’ problem-solving abilities; they can now apply the process with their partners to the problem being solved.

• Use this technique with a multi-step problem. Model using a starter sentence such as: “The problem is asking … ,” “I can use the strategy of … ,” “The steps I will use in solving this problem are … ,” “This problem is similar to … ,” and “I can check my answer by … .”

• Use this technique for a variety of problem types. Listen for comprehension of skills, concepts, and procedures as well as precision of language.

Applications• Print and frame sizes for photographs are explored in Exercise 37 on page 213. Standard

photographic sizes are: 2.5” × 3.5”, 3.5” × 5”, 4” × 6”, 5” × 7”, 8” × 10”, 11” × 14”, and 16” × 20”. Ask students whether any of these could be a dilation of another.

Pacing Suggestion• The formal lesson is long; however, experience with the explorations helps students develop

essential understanding of dilations. You might consider having half the class do Exploration 1(a) and the other half do Exploration 1(b).


Laurie’s Notes

T-207 Chapter 4

ExplorationMotivate• Cut a rectangle out of heavier card stock. Use a flashlight to cast a shadow of the rectangle

onto the wall. “Do the angles still appear to be right angles?” yes; They should be unless the rectangle is not parallel to the surface it is reflected onto.

• Vary the distance between the bulb of the flashlight and the rectangle. Discuss how this changes the shadow.

“Is the shadow always similar to the original figure?” It should be when the figure is held parallel to the wall and the flashlight is perpendicular to the wall.

Exploration 1• This first exploration serves to familiarize students with the dilate command in the dynamic

geometry software. Students should also be comfortable with the exploration posed verbally so that they are not looking at the result of the dilation in the book.

• Students should be comfortable clicking and dragging to change the shape and location of △ABC.

Popsicle Sticks: “What do you observe about the side lengths?” The sides of △ A′B′C′ are twice the length of the sides of △ ABC. “What do you observe about the angle measures?” The measures of corresponding angles are equal.

• In part (b), students should again try different locations and shapes for △ ABC and make observations about the coordinates of △ A′B′C′.

• Turn and Talk: Discuss observations with partners when the center is (0, 0) and the scale factor is 1 — 2 . Extension: “When the scale factor is 1, what will the image look like?” It will be congruent to the preimage.

• Extension: Usually there is at least one student who will enter a negative scale factor such as − 1 — 2 . If so, then the student should share his or her observations with the class. If time permits, have all students try a negative scale factor. Explain the effect of the negative sign.

Exploration 2• You could pose this exploration to students verbally so that they are not looking at the result

of the dilation in the book.• Ask students to make a conjecture about what they think is going to happen when they dilate

a line through the origin and a line not through the origin.• When students dilate the line through the origin, they may say, “Nothing happened!” Do not

give it away. Have them dilate a line not through the origin, which they will be able to see.• Big Idea: A line through the origin has an equation y = kx (direct variation). Multiplying

(x, y) by a scale factor of 3 will result in ordered pairs that are on the same line. (3y = 3kx simplifies to y = kx.) A line that does not pass through the origin has an equation y = mx + b, where b ≠ 0. Multiplying (x, y) by a scale factor of 3 will result in ordered pairs that are on a line parallel to the original line.

Communicate Your AnswerCommunicate Your Answer• Ask different students to share their understanding of what it means to dilate a figure.

Connecting to Next Step• Students have now been introduced to dilations. In the formal lesson, students will construct

dilations with a compass and a straightedge and will construct dilations in the coordinate plane.


HSG-CO.A.2 Represent transformations in the plane using, e.g., . . . geometry software; . . . Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

HSG-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.












Section 4.5 207

Section 4.5 Dilations 207

Dilations4.5

Dilating a Triangle in a Coordinate Plane

Work with a partner. Use dynamic geometry software to draw any triangle and label

it △ABC.

a. Dilate △ABC using a scale factor of 2 and a center of dilation at the origin to form

△A′B′C′. Compare the coordinates, side lengths, and angle measures of △ABC

and △A′B′C′.

DA

B

CA′

B′

C′

0

3

2

1

4

5

6

0 1 2 3 4 5 6 7 8

SamplePointsA(2, 1)B(1, 3)C(3, 2)SegmentsAB = 2.24BC = 2.24AC = 1.41Anglesm∠A = 71.57°m∠B = 36.87°m∠C = 71.57°

b. Repeat part (a) using a scale factor of 1 —

2 .

c. What do the results of parts (a) and (b) suggest about the coordinates, side lengths,

and angle measures of the image of △ABC after a dilation with a scale factor of k?

Dilating Lines in a Coordinate Plane

Work with a partner. Use dynamic geometry software to draw �� AB that passes

through the origin and �� AC that does not pass through the origin.

a. Dilate �� AB using a scale factor

of 3 and a center of dilation at

the origin. Describe the image.

b. Dilate �� AC using a scale factor

of 3 and a center of dilation at

the origin. Describe the image.

c. Repeat parts (a) and (b) using a

scale factor of 1 —

4 .

d. What do you notice about

dilations of lines passing

through the center of dilation

and dilations of lines not passing

through the center of dilation?

Communicate Your AnswerCommunicate Your Answer 3. What does it mean to dilate a fi gure?

4. Repeat Exploration 1 using a center of dilation at a point other than the origin.

Essential QuestionEssential Question What does it mean to dilate a fi gure?COMMON

CORE

Learning StandardsHSG-CO.A.2HSG-SRT.A.1aHSG-SRT.A.1b

A

CB

1

0

2

−1−2−3

−1

−2

0 1 2 3

LOOKING FOR STRUCTURE

To be profi cient in math, you need to look closely to discern a pattern or structure.

PointsA(−2, 2)B(0, 0)C(2, 0)

Linesx + y = 0x + 2y = 2

Sample

HSCC_GEOM_PE_04.05.indd 207 2/24/14 10:29 AM 2. a. The image is a line that coincides

with �� AB .

b. The image is a line that is parallel to �� AC . The x- and y-intercepts of the image are

each three times the x- and y-intercepts

of �� AC .

c. The image of �� AB is a line that coincides

with �� AB . The image of �� AC is a line that

is parallel to �� AC . The x- and y-intercepts

of the image are each one-fourth of the

x- and y-intercepts of �� AC .

d. When you dilate an image that passes

through the center of dilation, the

image coincides with the preimage.

When you dilate a line that does not

pass through the center of dilation, the

image is parallel to the preimage, and the

image has intercepts that can be found

by multiplying the intercepts of the

preimage by the constant of dilation.

3. to reduce or enlarge a fi gure so that the

image is proportional to the preimage


ANSWERS1. a. Check students’ work. The

x-value of each vertex of

△A′B′C′ is twice the x-value

of its corresponding vertex of

△ABC, and the y-value of each

vertex of △A′B′C′ is twice the

y-value of its corresponding

vertex of △ABC. Each side of

△A′B′C′ is twice as long as its

corresponding side of △ABC.

Each angle of △A′B′C′ is

congruent to its corresponding

angle of △ABC.

b. Sample answer:

The x-value of each vertex of

△A′B′C′ is half of the x-value



vertex of △A′B′C′ is half of

the y-value of its corresponding


△A′B′C′ is half as long as its

corresponding side of △ABC.



angle of △ABC.

c. The x-value of each vertex of

△A′B′C′ is k times the x-value



vertex of △A′B′C′ is k times

the y-value of its corresponding


△A′B′C′ is k times as long as

its corresponding side of △ABC.



angle of △ABC.

x

2

3

4

1

3 41 2

y

C

D A′

B

A

B′C′


208 Chapter 4


4.5 Lesson What You Will LearnWhat You Will Learn Identify and perform dilations.

Solve real-life problems involving scale factors and dilations.

Identifying and Performing Dilations

Identifying Dilations

Find the scale factor of the dilation. Then tell whether the dilation is a reduction or

an enlargement.

a.

C

PP′

128

b.

C

P

P′ 30

18

SOLUTION

a. Because CP′ — CP

= 12

— 8

, the scale factor is k = 3 —

2 . So, the dilation is an enlargement.

b. Because CP′ — CP

= 18

— 30

, the scale factor is k = 3 —

5 . So, the dilation is a reduction.


1. In a dilation, CP′ = 3 and CP = 12. Find the scale factor. Then tell whether the

dilation is a reduction or an enlargement.

READINGThe scale factor of a dilation can be written as a fraction, decimal, or percent.

dilation, p. 208center of dilation, p. 208scale factor, p. 208enlargement, p. 208reduction, p. 208


Core Core ConceptConceptDilationsA dilation is a transformation in which a fi gure is enlarged or reduced with respect

to a fi xed point C called the center of dilation and a scale factor k, which is the

ratio of the lengths of the corresponding sides of the image and the preimage.

A dilation with center of dilation C and scale factor k maps every point P in a

fi gure to a point P′ so that the following are true.

• If P is the center point C, then P = P′.

• If P is not the center point C, then the image

point P′ lies on �� CP . The scale factor k is a

positive number such that k = CP′ — CP

.

• Angle measures are preserved.

QC

P

R

P′

Q′

R′

A dilation does not change any line that passes through the center of dilation. A

dilation maps a line that does not pass through the center of dilation to a parallel line.

In the fi gure above, �� PR � �� P′R′ , �� PQ � �� P′Q′ , and �� QR � �� Q′R′ .

When the scale factor k > 1, a dilation is an enlargement. When 0 < k < 1, a dilation

is a reduction.


Extra Example 1Find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.

a.

C

P

P′12

4

k = 1 — 3 ; reduction

b.

CP

P′ 25

10

k = 5 — 2 ; enlargement

Teacher ActionsTeacher ActionsLaurie’s Notes• Take time to fully discuss the definition of dilation. Connect to the explorations completed by

students. “What will the scale factor k tell you about the dilation?” If students have done the explorations, they will know that when k > 1, the dilation will be an enlargement and when 0 < k < 1, the dilation will be a reduction.

• Students may have tried a scale factor that was negative with the dynamic geometry software. This is discussed on page 210.

• Turn and Talk: Have students discuss how to find the scale factor for each dilation.


AuditorySome students may have difficulty distinguishing dilations from rigid transformations. Ask students for examples of dilations in the real world, and have them explain why these are dilations. Answers may include enlarging a photo, changing the size of a font on a computer, and zooming on a digital image. Ensure student explanations mention that the size has changed, but relative positions in a figure have not.

MONITORING PROGRESS ANSWER 1. k =

1 —

4 ; reduction


Section 4.5 209


Dilating a Figure in the Coordinate Plane

Graph △ABC with vertices A(2, 1), B(4, 1), and C(4, −1) and its image after a

dilation with a scale factor of 2.

SOLUTIONUse the coordinate rule for a dilation with

k = 2 to fi nd the coordinates of the vertices

of the image. Then graph △ABC and its image.

(x, y) → (2x, 2y)

A(2, 1) → A′(4, 2)

B(4, 1) → B′(8, 2)

C(4, −1) → C′(8, −2)

Notice the relationships between the lengths and slopes of the sides of the triangles in

Example 2. Each side length of △A′B′C′ is longer than its corresponding side by the

scale factor. The corresponding sides are parallel because their slopes are the same.

Core Core ConceptConceptCoordinate Rule for DilationsIf P(x, y) is the preimage of a point, then its image

after a dilation centered at the origin (0, 0) with

scale factor k is the point P′(kx, ky).READING DIAGRAMSIn this chapter, for all of the dilations in the coordinate plane, the center of dilation is the origin unless otherwise noted.

x

y P′(kx, ky)

P(x, y)

x

y

2

−2

2 6

A

CC′

B′A′

B

Dilating a Figure in the Coordinate Plane

Graph quadrilateral KLMN with vertices K(−3, 6), L(0, 6), M(3, 3), and N(−3, −3)

and its image after a dilation with a scale factor of 1 —

3 .

SOLUTIONUse the coordinate rule for a dilation with k =

1 —

3 to

fi nd the coordinates of the vertices of the image.

Then graph quadrilateral KLMN and its image.

(x, y) → ( 1 — 3 x, 1 — 3 y ) K(−3, 6) → K′(−1, 2)

L(0, 6) → L′(0, 2)

M(3, 3) → M′(1, 1)

N(−3, −3) → N′(−1, −1)


Graph △PQR and its image after a dilation with scale factor k.

2. P(−2, −1), Q(−1, 0), R(0, −1); k = 4

3. P(5, −5), Q(10, −5), R(10, 5); k = 0.4

x

y

4

42−2−4

K L

M

N

K′ L′

M′

N′

(x, y) → (kx, ky)


Extra Example 2Graph △PQR with vertices P(0, 2), Q(1, 0), and R(2, 2) and its image after a dilation with scale factor 3.

x

y

4

64Q′

R′P′

Q

P R

Extra Example 3Graph △PQR with vertices P(4, 6), Q(−4, 2), and R(2, −6) and its image after a dilation with scale factor 0.5.

x

y6

−4

−6

42−4

Q′

R′

P′Q

P

R


ComprehensionMake sure students understand the relationship between the scale factor and the effects of a dilation. Use examples to guide students through a discussion developing the inequalities 0 < k < 1 and k < 0 as guides for identifying whether a dilation with a scale factor k is a reduction or an enlargement.

Teacher ActionsTeacher ActionsLaurie’s Notes• Write the Core Concept.

Fact-First Question: “If the preimage is in Quadrant III, then the image will be in Quadrant III. Explain why.” The point P′ is on �� CP , where C is the center of dilation, so P′ will still be in Quadrant III.

• Think-Alouds: Pose Example 2 and say, “To solve this example, I need to … .” Ask partner A to think aloud for partner B to hear the problem-solving process. When students have finished the example, Popsicle Sticks a response.

• Think-Alouds: Pose Example 3 and say, “In this example, I predict … .” Ask partner B to think aloud for partner A to hear the prediction. When students have finished the example, Popsicle Sticks a response.



x−6−8

y

−2P

P′

Q′

R′

Q

R


210 Chapter 4


In the coordinate plane, you can have scale factors that are negative numbers. When

this occurs, the fi gure rotates 180°. So, when k > 0, a dilation with a scale factor of

−k is the same as the composition of a dilation with a scale factor of k followed by a

rotation of 180° about the center of dilation. Using the coordinate rules for a dilation

and a rotation of 180°, you can think of the notation as

(x, y) → (kx, ky) → (−kx, −ky).

Using a Negative Scale Factor

Graph △FGH with vertices F(−4, −2), G(−2, 4), and H(−2, −2) and its image

after a dilation with a scale factor of − 1 — 2 .

SOLUTIONUse the coordinate rule for a dilation with k = − 1 —

2 to fi nd the coordinates of the

vertices of the image. Then graph △FGH and its image.

(x, y) → ( − 1 — 2 x, − 1 — 2 y ) F(−4, −2) → F′(2, 1)

G(−2, 4) → G′(1, −2)

H(−2, −2) → H′(1, 1)


4. Graph △PQR with vertices P(1, 2), Q(3, 1), and R(1, −3) and its image after a

dilation with a scale factor of −2.

5. Suppose a fi gure containing the origin is dilated. Explain why the corresponding

point in the image of the fi gure is also the origin.

center ofdilation

preimage

scale factor k

scale factor −k

x

y

G

HF

F′H′

G′

x

y4

2

−4

−2

42−4

Step 1 Step 2 Step 3

Q

C

P

R

Q

C

P

R

P′

Q′

R′

Q

C

P

R

P′

Q′

R′

Draw a triangle Draw △PQR and

choose the center of the dilation C

outside the triangle. Draw rays from

C through the vertices of the triangle.

Use a compass Use a compass to

locate P′ on �� CP so that CP′ = 2(CP).

Locate Q′ and R′ using the same

method.

Connect points Connect points

P′, Q′, and R′ to form △P′Q′R′.

Constructing a Dilation

Use a compass and straightedge to construct a dilation of △PQR with a scale factor

of 2. Use a point C outside the triangle as the center of dilation.

SOLUTION


Extra Example 4Graph △FGH with vertices F(3, 6), G(3, −3), and H(6, 6) and its image after a dilation with a scale factor of − 1 — 3 .

x

y

4

6

62−2

G′

H′ F′G

HF

Teacher ActionsTeacher ActionsLaurie’s Notes• The construction of a dilation with scale factor of 2 is shown. You

could differentiate at this point and hand slips of paper to different groups stating the scale factor. Groups of students ready for a more demanding task could use scale factors of 5 — 2 , 3, 1 — 2 , or 2 — 3 .

• MP5 Use Appropriate Tools Strategically: Use dynamic geometry software to demonstrate negative scale factors, or search the Internet (“GCSE transformations videos”) for a video. Using a scalene figure will help students see the rotation of 180°.

• Connection: Rotation of 180° about the origin maps (x, y) → (−x, −y). Dilation by scale factor of −k centered at the origin maps (x, y) → (−kx, −ky).

Always-Sometimes-Never True: “A dilation of −k results in an image in Quadrant III.” sometimes true; When the preimage is in Quadrant I, the image will be in Quadrant III; otherwise it will be in a different quadrant.

• Circulate as students try Example 4 with their partners.


4.

5. According to the Coordinate Rule for

Dilations, if the origin P(0, 0) is the

preimage of a point, then its image

after a dilation centered at the origin

with a scale factor k is the point

P′(k0, k0), which is also the origin,

or (0, 0).

x

4

6

2

−4−6

y

−2

−4

P

P′

Q′

R′

Q

R


Section 4.5 211



Finding a Scale Factor

You are making your own photo stickers.

Your photo is 4 inches by 4 inches. The image

on the stickers is 1.1 inches by 1.1 inches.

What is the scale factor of this dilation?

SOLUTIONThe scale factor is the ratio of a side

length of the sticker image to a side

length of the original photo, or 1.1 in.

— 4 in.

.

So, in simplest form, the scale factor is 11

— 40

.

Finding the Length of an Image

You are using a magnifying glass that shows the

image of an object that is six times the object’s

actual size. Determine the length of the image

of the spider seen through the magnifying glass.

SOLUTION

image length

—— actual length

= k

x —

1.5 = 6

x = 9

So, the image length through the magnifying glass is 9 centimeters.


6. An optometrist dilates the pupils of a patient’s eyes to get a better look at the back

of the eyes. A pupil dilates from 4.5 millimeters to 8 millimeters. What is the scale

factor of this dilation?

7. The image of a spider seen through the magnifying glass in Example 6 is shown at

the left. Find the actual length of the spider.

When a transformation, such as a dilation, changes the shape or size of a fi gure, the

transformation is nonrigid. In addition to dilations, there are many possible nonrigid

transformations. Two examples are shown below. It is important to pay close attention

to whether a nonrigid transformation preserves lengths and angle measures.

Horizontal Stretch Vertical Stretch

A

C B B′

A

C B

A′

READINGScale factors are written so that the units in the numerator and denominator divide out.

4 in.

1.1 in.

1.5 cm

12.6 cm


Extra Example 5You are using word processing software to type the online school newsletter. You change the size of the text in one headline from 0.5 inch tall to 1.25 inches tall. What is the scale factor of this dilation? 2.5

Extra Example 6You are using a magnifying glass that shows the image of an object that is six times the object’s actual size. The image of a spider seen through the magnifying glass is 13.5 centimeters long. Find the actual length of the spider. 2.25 centimeters

• Help students understand that the original photo is the preimage. The image is smaller, meaning this is a reduction.

• Students may be confused because there is no “center” specified. The scale factor is the ratio of corresponding sides comparing preimage side length to image side length.

• Look Back: Give students time to reflect on this lesson. Previous transformations in the chapter were rigid and dilations are not. The result of a dilation is an image that is the same shape but

not necessarily the same size. A transformation might distort the shape as shown at the bottom of the page. A stretch horizontally or vertically does not preserve length or angle measure.

ClosureClosure• Writing Prompt: How are dilations alike/different from rigid

transformations? Sample answer: They are alike because they both preserve angle measure. They are different because the lengths of the sides are congruent for rigid transformations and proportional for dilations.

Teacher ActionsTeacher ActionsLaurie’s Notes


6. 16 —

9

7. 2.1 cm


212 Chapter 4



1. COMPLETE THE SENTENCE If P(x, y) is the preimage of a point, then its image after a dilation

centered at the origin (0, 0) with scale factor k is the point ___________.

2. WHICH ONE DOESN’T BELONG? Which scale factor does not belong with the other three? Explain

your reasoning.

5 —

4 115% 260%


In Exercises 3–6, fi nd the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. (See Example 1.)

3.

C

PP′

14

6

4.

CP

P′ 24

9

5.

C

P

P′15

9

6.

C

P

P′

828

CONSTRUCTION In Exercises 7–10, copy the diagram. Then use a compass and straightedge to construct a dilation of △LMN with the given center and scale factor k.

M

L

N

P

C

7. Center C, k = 2

8. Center P, k = 3

9. Center M, k = 1 —

2

10. Center C, k = 25%

CONSTRUCTION In Exercises 11–14, copy the diagram. Then use a compass and straightedge to construct a dilation of quadrilateral RSTU with the given center and scale factor k.

U

R

S

T

P

C

11. Center C, k = 3 12. Center P, k = 1 —

3

13. Center R, k = 0.25 14. Center C, k = 75%

In Exercises 15–18, graph the polygon and its image after a dilation with scale factor k. (See Examples 2 and 3.)

15. X(6, −1), Y(−2, −4), Z(1, 2); k = 3

16. A(0, 5), B(−10, −5), C(5, −5); k = 120%

17. T(9, −3), U(6, 0), V(3, 9), W(0, 0); k = 2 —

3

18. J(4, 0), K(−8, 4), L(0, −4), M(12, −8); k = 0.25

In Exercises 19–22, graph the polygon and its image after a dilation with scale factor k. (See Example 4.)

19. B(−5, −10), C(−10, 15), D(0, 5); k = − 1 — 5

20. L(0, 0), M(−4, 1), N(−3, −6); k = −3

21. R(−7, −1), S(2, 5), T(−2, −3), U(−3, −3); k = −4

22. W(8, −2), X(6, 0), Y(−6, 4), Z(−2, 2); k = −0.5



ANSWERS1. P′(kx, ky)

2. 60%; Because 0.6 < 1, 60% is a scale

factor for a reduction. The other three

are scale factors for enlargements.

3. 3—7 ; reduction

4. 8—3 ; enlargement

5. 3—5 ; reduction

6. 7—2 ; enlargement


15.

16.

x

4

4 8 12 16−4

y

−8

−12

Z

Y′

Z′

X′X

Y

x

2

6

4−4−8−12

y

−4

A

A′

B′ C′

B C


ASSIGNMENT

Basic: 1, 2, 3–35 odd, 38, 48, 52–57

Average: 1, 2, 4, 8, 14, 19, 24–34 even, 38–42, 48, 52–57

Advanced: 1, 2, 10, 16, 22–36 even, 38–49, 52–57

HOMEWORK CHECK

Basic: 5, 7, 21, 25, 31

Average: 14, 19, 24, 26, 39

Advanced: 22, 30, 39, 46, 47

17.

x

4

6

8

2

4 6 8

y

−2

UW

W′ U′

T′

V′

V

T

18.


x

4

4 8 12−4−8

y

−8

K

K′J′

M′L′

J

L

M










Section 4.5 213


ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in fi nding the scale factor of the dilation.

23.

k = 12 — 3

= 4

✗

24.

k = 2 — 4

= 1 — 2

✗

In Exercises 25–28, the red fi gure is the image of the blue fi gure after a dilation with center C. Find the scale factor of the dilation. Then fi nd the value of the variable.

25.

C

35

9 15x

26.

C

28

12

14

n

27.

C

2 2y 3

28. C

74 m

28

29. FINDING A SCALE FACTOR You receive wallet-sized

photos of your school picture. The photo is

2.5 inches by 3.5 inches. You decide to dilate the

photo to 5 inches by 7 inches at the store. What is the

scale factor of this dilation? (See Example 5.)

30. FINDING A SCALE FACTOR Your visually impaired

friend asked you to enlarge your notes from class so

he can study. You took notes on 8.5-inch by 11-inch

paper. The enlarged copy has a smaller side with a

length of 10 inches. What is the scale factor of this

dilation? (See Example 5.)

In Exercises 31–34, you are using a magnifying glass. Use the length of the insect and the magnifi cation level to determine the length of the image seen through the magnifying glass. (See Example 6.)

31. emperor moth 32. ladybug

Magnifi cation: 5× Magnifi cation: 10×

60 mm 4.5 mm

33. dragonfl y 34. carpenter ant

Magnifi cation: 20× Magnifi cation: 15×

47 mm

12 mm12 mm

35. ANALYZING RELATIONSHIPS Use the given actual

and magnifi ed lengths to determine which of the

following insects were looked at using the same

magnifying glass. Explain your reasoning.

grasshopper black beetle

Actual: 2 in. Actual: 0.6 in.

Magnifi ed: 15 in. Magnifi ed: 4.2 in.

honeybee monarch butterfl y

Actual: 5 —

8 in. Actual: 3.9 in.

Magnifi ed: 75

— 16

in. Magnifi ed: 29.25 in.

36. THOUGHT PROVOKING Draw △ABC and △A′B′C′ so

that △A′B′C′ is a dilation of △ABC. Find the center

of dilation and explain how you found it.

37. REASONING Your friend prints a 4-inch by 6-inch

photo for you from the school dance. All you have is

an 8-inch by 10-inch frame. Can you dilate the photo

to fi t the frame? Explain your reasoning.

C

P

P′12

3

x

y4

−4

−6

P′(−4, 2)P(−2, 1)

4

21

2


ANSWERS23. The scale factor should be calculated

by fi nding CP′ — CP

, not CP

— CP′

; k = 3 —

12 =

1 —

4

24. The x-values (or y-values) of

corresponding vertices should

be used, not the lengths of

corresponding sides; k = 4 —

2 = 2

25. k = 5 —

3 ; x = 21

26. k = 2; n = 6

27. k = 2 —

3 ; y = 3

28. k = 1 —

4 ; m = 16

29. k = 2

30. k = 20

— 17

31. 300 mm

32. 45 mm

33. 940 mm

34. 180 mm

35. grasshopper, honey bee, and monarch

butterfl y; The scale factor for these

three is k = 15

— 2 . The scale factor for the

black beetle is k = 7.

36. Sample answer:

the origin (0,0); After drawing

△ABC and its dilation, draw the

lines connecting each vertex in the

preimage with its corresponding

vertex in the image. These three lines

intersect at the center of dilation.

37. no; The scale factor for the shorter

sides is 8 —

4 = 2, but the scale factor for

the longer sides is 10

— 6 =

5 —

3 . The scale

factor for both sides has to be the

same or the picture will be distorted.

x6 82 4−4 −2

y

−2

−6

−9

A

A′ B′

C′

B

C

(0, 0), (center of dilation)


214 Chapter 4


38. HOW DO YOU SEE IT? Point C is the center of

dilation of the images. The scale factor is 1 —

3 . Which

fi gure is the original fi gure? Which fi gure is the

dilated fi gure? Explain your reasoning.

C

39. MATHEMATICAL CONNECTIONS The larger triangle

is a dilation of the smaller triangle. Find the values of

x and y.

C2

6

x + 1

2x + 8

(3y − 34)°

(y + 16)°

40. WRITING Explain why a scale factor of 2 is the same

as 200%.

In Exercises 41– 44, determine whether the dilated fi gure or the original fi gure is closer to the center of dilation. Use the given location of the center of dilation and scale factor k.

41. Center of dilation: inside the fi gure; k = 3

42. Center of dilation: inside the fi gure; k = 1 —

2

43. Center of dilation: outside the fi gure; k = 120%

44. Center of dilation: outside the fi gure; k = 0.1

45. ANALYZING RELATIONSHIPS Dilate the line through

O(0, 0) and A(1, 2) using a scale factor of 2.

a. What do you notice about the lengths of — O′A′

and — OA ?

b. What do you notice about �� O′A′ and �� OA ?

46. ANALYZING RELATIONSHIPS Dilate the line through

A(0, 1) and B(1, 2) using a scale factor of 1 —

2 .

a. What do you notice about the lengths of — A′B′

and — AB ?

b. What do you notice about �� A′B′ and �� AB ?

47. ATTENDING TO PRECISION You are making a

blueprint of your house. You measure the lengths of

the walls of your room to be 11 feet by 12 feet. When

you draw your room on the blueprint, the lengths

of the walls are 8.25 inches by 9 inches. What scale

factor dilates your room to the blueprint?

48. MAKING AN ARGUMENT Your friend claims that

dilating a fi gure by 1 is the same as dilating a fi gure

by −1 because the original fi gure will not be

enlarged or reduced. Is your friend correct? Explain

your reasoning.

49. USING STRUCTURE Rectangle WXYZ has vertices

W(−3, −1), X(−3, 3), Y(5, 3), and Z(5, −1).

a. Find the perimeter and area of the rectangle.

b. Dilate the rectangle using a scale factor of 3. Find

the perimeter and area of the dilated rectangle.

Compare with the original rectangle. What do

you notice?

c. Repeat part (b) using a scale factor of 1 —

4 .

d. Make a conjecture for how the perimeter and area

change when a fi gure is dilated.

50. REASONING You put a reduction of a page on the

original page. Explain why there is a point that is in

the same place on both pages.

51. REASONING △ABC has vertices A(4, 2), B(4, 6), and

C(7, 2). Find the coordinates of the vertices of the

image after a dilation with center (4, 0) and a scale

factor of 2.

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyThe vertices of △ABC are A(2, −1), B(0, 4), and C(−3, 5). Find the coordinates of the vertices of the image after the translation. (Section 4.1)

52. (x, y) → (x, y − 4) 53. (x, y) → (x − 1, y + 3) 54. (x, y) → (x + 3, y − 1)

55. (x, y) → (x − 2, y) 56. (x, y) → (x + 1, y − 2) 57. (x, y) → (x − 3, y + 1)



ANSWERS38. larger star; smaller star; Because the

scale factor is between 0 and 1, the

dilation is a reduction.

39. x = 5, y = 25

40. A fi gure that is 200% larger than the

preimage will be twice as large.

41. original

42. dilated

43. original

44. dilated


50. The center of dilation must be on

that page. So, this point will be in

the same place for both the original

fi gure and the dilated fi gure.

51. A′ ( 4, 4 ) , B′ ( 4, 12 ) , C′ ( 10, 4 )

52. A′ ( 2, −5 ) , B′ ( 0, 0 ) , C′ ( −3, 1 )

53. A′ ( 1, 2 ) , B′ ( −1, 7 ) , C′ ( −4, 8 )

54. A′ ( −5, 2 ) , B′ ( 3, 3 ) , C′ ( 0, 4 )

55. A′ ( 0, −1 ) , B′ ( −2, 4 ) , C′ ( −5, 5 )

56. A′ ( 3, −3 ) , B′ ( 1, 2 ) , C′ ( −2, 3 )

57. A′ ( −1, 0 ) , B′ ( −3, 5 ) , C′ ( −6, 6 )

Mini-Assessment

1. Find the scale factor of the dilation. Then tell whether it is a reduction or an enlargement.

C

P

P′24

10

k = 5 — 12 , reduction

2. Graph △DEF with vertices D(2, 6), E(2, 2), and F(4, 2) and its image after a dilation with scale factor − 1 — 2 .

x

y

4

6

2

42

D′

E′F′

E F

D

3. A photographer enlarges a 4 inch × 5 inch photo to an 8 inch × 10 inch photo. What is the scale factor of the dilation? 2


















Section 4.6 T-214

Laurie’s Notes

Overview of Section 4.6Introduction• In grade 8, similar figures were defined as figures that have the same shape but not

necessarily the same size. Two figures are similar when corresponding side lengths are proportional and corresponding angles are congruent. Finally, perimeters and areas of similar figures and surface areas and volumes of similar solids were studied.

• In this section, similarity is defined in terms of similarity transformations. Students need to understand that a similarity transformation is a dilation or a composition of rigid motions and dilations. To decide (prove) whether two figures are similar, you need to map one figure onto the other by a single dilation or a composition of rigid motions and dilations. The composition of rigid motions and dilations may not be unique.

• FYI: Similarity is revisited in Chapter 8, where different conditions are presented to show that triangles are similar. The transformational approach is presented in this lesson.

Resources• Use dynamic geometry software to demonstrate, or perform, a similarity transformation. Use

the software to determine whether there is a similarity transformation that maps one figure onto another.

Formative Assessment Tips• 3-2-1: This formative assessment strategy is useful in giving students a structured way in

which to reflect on their learning, particularly at the conclusion of a unit of study. Students are asked to respond to three writing prompts, giving three responses to the first prompt, two responses to the second prompt, and one response to the third prompt. All six responses relate to what students have learned during the unit of study—in this case, a chapter on transformations.

• Distribute a printed 3-2-1 reflection sheet and give time for students to reflect on their learning and to write.• 3 new things (concepts, skills, procedures, … ) I learned in this chapter• 2 things (concepts, skills, procedures, … ) I am still struggling with• 1 thing that will help me tomorrow

• Collect and review student reflections and plan instruction for tomorrow accordingly. For students, the reflection allows them to see what learning has occurred and where they should focus their attention.

Pacing Suggestion• Once students have worked the explorations, continue with the formal lesson. Alternatively,

simply use the software to perform the similarity transformations in the formal lesson.


Laurie’s Notes

T-215 Chapter 4

ExplorationMotivate• Draw a simple stick figure or other image on a stretchable surface, such as a balloon, physical

therapy elastic, or play putty. Ask students what they think will happen to the figure when you pull the picture to the right. The image will be distorted. Pull one of the sides of the picture to confirm.

• Pull the top of the picture so that students see this result as the same.Ask students what they think will happen when the stretchable surface is pulled in both directions (right and up). The image will enlarge proportionally.

• Alternate Approach: If you can display a computer image to the class, you can drag a side to distort the image, or drag a corner to change the size of the image proportionally.

Discuss• Similar figures can be congruent. Two congruent figures meet the definition of similarity:

corresponding side lengths are proportional (1 : 1) and corresponding angles are congruent.• Note that the definition of a dilation excludes a scale factor of 1. For that reason, the dilation

is referred to as a nonrigid motion because the size changes.

Exploration 1• Note that the center of dilation does not need to be the origin.

MP3 Construct Viable Arguments and Critique the Reasoning of Others: “Are the triangles similar? How do you know?” Listen for the ratio of corresponding sides all equal 3, and the corresponding angles are congruent.

• Extension: Have students repeat the exploration (a) using one of the vertices as the center of dilation and (b) using a point inside the original triangle (but not the origin) as the center of dilation.

“When the center of dilation is a vertex of the original triangle, will the triangles be similar? Explain.” yes; The ratio of corresponding sides all equal 3, and the corresponding angles are congruent.

“When the center of dilation is inside the original triangle, will the triangles be similar? Explain.” yes; The ratio of corresponding sides all equal 3, and the corresponding angles are congruent.

Exploration 2• In each construction, ask students how they know the preimage and the image are similar.• MP2 Reason Abstractly and Quantitatively and MP3: What you hope is that students

will say that translations, reflections, and rotations are isometries, congruence-preserving transformations. If two figures are congruent, then they are similar. They should not need to measure!Always-Sometimes-Never True: “If two figures are congruent, then they are similar.”always trueAlways-Sometimes-Never True: “If two figures are similar, then they are congruent.” sometimes true

Communicate Your AnswerCommunicate Your Answer• Listen for student understanding of rigid transformations producing congruent and, hence,

similar images. Dilations are nonrigid, and the image is similar to the original figure.• Question 4 prepares students for content in the formal lesson.

Connecting to Next Step• The explorations should be a review of rigid and nonrigid transformations as well as the

definition of similar figures. Quickly transition to the formal lesson.


HSG-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, … or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

HSG-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.












Section 4.6 215

Section 4.6 Similarity and Transformations 215

Similarity and Transformations4.6

Dilations and Similarity



b. Dilate the triangle using a scale factor of 3. Is the image similar to the original

triangle? Justify your answer.

0

1

2

3

−1

−1

−2

−3

−2−3−4−5−6 0 1 2

A

DC

B

A′

C′

B′

3

BB

SamplePointsA(−2, 1)B(−1, −1)C(1, 0)D(0, 0)SegmentsAB = 2.24BC = 2.24AC = 3.16Anglesm∠A = 45°m∠B = 90°m∠C = 45°

Rigid Motions and Similarity


a. Use dynamic geometry software to draw any triangle.

b. Copy the triangle and translate it 3 units left and 4 units up. Is the image similar to

the original triangle? Justify your answer.

c. Refl ect the triangle in the y-axis. Is the image similar to the original triangle?

Justify your answer.

d. Rotate the original triangle 90° counterclockwise about the origin. Is the image

similar to the original triangle? Justify your answer.

Communicate Your AnswerCommunicate Your Answer 3. When a fi gure is translated, refl ected, rotated, or dilated in the plane, is the image

always similar to the original fi gure? Explain your reasoning.

4. A fi gure undergoes a composition of transformations, which includes translations,

refl ections, rotations, and dilations. Is the image similar to the original fi gure?


ATTENDING TO PRECISION

To be profi cient in math, you need to use clear defi nitions in discussions with others and in your own reasoning.

Essential QuestionEssential Question When a fi gure is translated, refl ected, rotated,

or dilated in the plane, is the image always similar to the original fi gure?

Two fi gures are similar fi gures

when they have the same shape

but not necessarily the same size.

COMMON CORE

Learning StandardsHSG-CO.A.5HSG-SRT.A.2

A

CB

F

G

E

Similar Triangles



b. Check students’ work; yes; Each

side of △A′B′C′ is three times

as long as its corresponding side

of △ABC. The corresponding

angles are congruent. Because

the corresponding sides are

proportional and the corresponding

angles are congruent, the image is

similar to the original triangle.

2. a. Sample answer:

b. Sample answer:

yes; Because the corresponding

sides are congruent and the

corresponding angles are

congruent, the image is similar to

the original triangle.

2c–d. See Additional Answers.

3. yes; The corresponding sides are

always congruent or proportional, and

the corresponding angles are always

congruent.

−2−3 −1 0

0

−2

−1

1

2

1 2 3

A

B

C

−4−5 −3 −2

0

−1

3

4

2

5

−1 0 1

1A

B

C

A′

B′

C′

4. yes; According to Composition Theorem

(Thm. 4.2), the composition of two or more

rigid motions is a rigid motion. Also, a

dilation preserves angle measures and results

in an image with lengths proportional to the

preimage lengths. So, a composition of rigid

motions or dilations will result in an image

that has angle measures congruent to the

corresponding angle measures of the original

fi gure, and sides that are either congruent or

proportional to the corresponding sides of

the original fi gure.


216 Chapter 4


4.6 Lesson What You Will LearnWhat You Will Learn Perform similarity transformations.

Describe similarity transformations.

Prove that fi gures are similar.

Performing Similarity TransformationsA dilation is a transformation that preserves shape but not size. So, a dilation is a

nonrigid motion. A similarity transformation is a dilation or a composition of rigid

motions and dilations. Two geometric fi gures are similar fi gures if and only if there is

a similarity transformation that maps one of the fi gures onto the other. Similar fi gures

have the same shape but not necessarily the same size.

Congruence transformations preserve length and angle measure. When the scale factor

of the dilation(s) is not equal to 1 or −1, similarity transformations preserve angle

measure only.

Performing a Similarity Transformation

Graph △ABC with vertices A(−4, 1), B(−2, 2), and C(−2, 1) and its image after the

similarity transformation.


Dilation: (x, y) → (2x, 2y)

SOLUTION

Step 1 Graph △ABC.

Step 2 Translate △ABC 5 units right and 1 unit up. △A′B′C′ has vertices

A′(1, 2), B′(3, 3), and C′(3, 2).

Step 3 Dilate △A′B′C′ using a scale factor of 2. △A″B″C ″ has endpoints

A″(2, 4), B″(6, 6), and C ″(6, 4).


1. Graph — CD with endpoints C(−2, 2) and D(2, 2) and its image after the



Dilation: (x, y) → ( 1 — 2 x,

1 —

2 y )

2. Graph △FGH with vertices F(1, 2), G(4, 4), and H(2, 0) and its image after the



Dilation: (x, y) → (1.5x, 1.5y)

similarity transformation, p. 216

similar fi gures, p. 216


x

y

4

2

8

6

42 86−2−4

B(−2, 2)A(−4, 1)

C(−2, 1)C′(3, 2)A′(1, 2)

B′(3, 3)

A″(2, 4)B″(6, 6)

C″(6, 4)


Extra Example 1Graph — AB with endpoints A(12, −6) and B(0, −3) and its image after the similarity transformation.Reflection: in the y-axis

Dilation: (x, y) → ( 1 — 3 x, 1 — 3 y )

x

y4

−8

8 12−8−12

A′(−12, −6)B′(0, −3)

A″(−4, −2)

B″(0, −1)

A(12, −6)

B(0, −3)

Teacher ActionsTeacher ActionsLaurie’s Notes• Discuss dilations—they are nonrigid, meaning that the size will change and the scale factor k

does not equal 1. Similar figures on the other hand can be congruent.• Note that similarity transformation is defined and then used to determine whether two figures

are similar. Again note the “if and only if” portion of the statement. To help demonstrate this, draw two similar trapezoids, rotating one 90°, on the board. The figures are similar if there are similarity transformations mapping one to the other, and if the figures are similar, then there are similarity transformations that would map one to the other.

• Pose Example 1 and ask for a Thumbs Up indication when partners are ready to begin the example.• Monitoring Progress: Using whiteboards, have half the class do Question 1 and the other

half do Question 2.


OrganizationHave students create a chart to compare the properties of congruence transformations and similarity transformations.

Transfor-mation

Congru-ence

Similarity (k ≠ 1, −1)

Preserves Angle Congru-ence

Preserves Length

Preserves Shape

Preserves Size


2.

x2

y

−2C′

C ″

D″

C DD′

x

4

2

4 6

y

−2

−4

−6

F

F′

H′

G′F″

G

H

G″

H″


Section 4.6 217


Describing Similarity Transformations

Describing a Similarity Transformation

Describe a similarity transformation that maps trapezoid PQRS to trapezoid WXYZ.

x

y4

2

−4

4 6−2−4

P Q

S R

W

ZY

X

SOLUTION — QR falls from left to right, and

— XY rises from left to right. If you refl ect

trapezoid PQRS in the y-axis as

shown, then the image, trapezoid

P′Q′R′S′, will have the same

orientation as trapezoid WXYZ.

Trapezoid WXYZ appears to be about one-third as large as trapezoid P′Q′R′S′. Dilate trapezoid P′Q′R′S′ using a scale factor of

1 —

3 .

(x, y) → ( 1 — 3 x, 1 — 3 y )

P′(6, 3) → P″(2, 1)

Q′(3, 3) → Q″(1, 1)

R′(0, −3) → R″(0, −1)

S′(6, −3) → S″(2, −1)

The vertices of trapezoid P″Q″R″S″ match the vertices of trapezoid WXYZ.

So, a similarity transformation that maps trapezoid PQRS to trapezoid WXYZ is a

refl ection in the y-axis followed by a dilation with a scale factor of 1 —

3 .


3. In Example 2, describe another similarity

transformation that maps trapezoid PQRS

to trapezoid WXYZ.

4. Describe a similarity transformation that maps

quadrilateral DEFG to quadrilateral STUV.

x

y4

2

4−2−4

W

ZY

X

P(−6, 3) P′(6, 3)Q′(3, 3)

S′(6, −3)R′(0, −3)

Q(−3, 3)

S(−6, −3) R(0, −3)

x

y4

2

−4

2−4

DE

FGU

TS

V


Extra Example 2Describe a similarity transformation that maps trapezoid WXYZ to trapezoid PQRS.

x

y4

−4

6

PQ

SRW

ZY

X

Sample answer: A reflection in the x-axis followed by a dilation with a scale factor of 2.

Teacher ActionsTeacher ActionsLaurie’s Notes• Describing a similarity transformation is more challenging than performing a similarity

transformation. Students are often uncertain where to begin.• Teaching Tip: Have students focus on a pair of corresponding sides. What dilation about the

origin will make them the same size? Does the orientation need to change (reflection)? Does the location need to change (translation)? Does the position need to change (rotation)?

• Alternate Approach: Have students use dynamic geometry software to answer the question.• Monitoring Progress: Question 3 suggests that the solution (similarity transformation) is not

unique! This is a Big Idea for students to explore.• MP5 Use Appropriate Tools Strategically: If students are using dynamic geometry

software, take time for them to generate a sample of different similarity transformations for Example 2.

MONITORING PROGRESS ANSWERS 3. Sample answer: refl ection in the

x-axis followed by a dilation with a

scale factor of − 1 —

3

4. Sample answer: dilation with a

scale factor of 1 —

2 followed by a 180°

rotation about the origin


218 Chapter 4


Proving Figures Are SimilarTo prove that two fi gures are similar, you must prove that a similarity transformation

maps one of the fi gures onto the other.

Proving That Two Squares Are Similar

Prove that square ABCD is similar to square EFGH.

Given Square ABCD with side length r,

square EFGH with side length s, — AD � — EH

Prove Square ABCD is similar to

square EFGH.

SOLUTIONTranslate square ABCD so that point A maps to point E. Because translations map

segments to parallel segments and — AD � — EH , the image of

— AD lies on — EH .

E H

GF

D′

B′ C′

rs

A E H

GF

D

CB

rs

Because translations preserve length and angle measure, the image of ABCD, EB′C′D′,is a square with side length r. Because all the interior angles of a square are right

angles, ∠B′ED′ ≅ ∠FEH. When ED′ coincides with EH , EB′ coincides with EF . So,

— EB′ lies on

— EF . Next, dilate square EB′C′D′ using center of dilation E. Choose the

scale factor to be the ratio of the side lengths of EFGH and EB′C′D′, which is s —

r .

E H

GF

D′

B′ C′

rs

E H

GF

s

This dilation maps — ED′ to

— EH and — EB′ to

— EF because the images of — ED′ and

— EB′

have side length s —

r (r) = s and the segments

— ED′ and — EB′ lie on lines passing through

the center of dilation. So, the dilation maps B′ to F and D′ to H. The image of C′ lies

s —

r (r) = s units to the right of the image of B′ and

s —

r (r) = s units above the image of D′.

So, the image of C′ is G.

A similarity transformation maps square ABCD to square EFGH. So,

square ABCD is similar to square EFGH.


5. Prove that △JKL is similar to △MNP.

Given Right isosceles △JKL with leg length t, right isosceles △MNP with leg

length v, — LJ � — PM

Prove △JKL is similar to △MNP.

A E H

GF

D

CB

rs

P

N

L

K

J

M

v

t


Extra Example 3Prove that square ABCD is similar to square EFGH.Given Square ABCD with side length s,

square EFGH with side length 2s, — AD || — EH

Prove Square ABCD is similar to square EFGH.

A D

C

F G

HE

B2s

s


Teacher ActionsTeacher ActionsLaurie’s Notes• Remind students of the definition of a square.

“How can you prove that two squares are similar?” You have to show that there is a similarity transformation that maps one square onto the other square.

• The proof may seem unnecessarily long and wordy to students, but it models what it means to prove that two figures are similar. We need to show that there is a sequence of similarity transformations that map square ABCD onto square EFGH. Note that if — AD and — EH were not given to be parallel, then a rotation would have been necessary as well.

Work through the proof as shown. Ask, “Is there another series of similarity transformations that map square ABCD onto square EFGH? Explain.” yes; One possibility would be to dilate first and then translate.

ClosureClosure• 3-2-1: Hand out a 3-2-1 reflection sheet as described on page T-214.

MONITORING PROGRESS ANSWER



Section 4.6 219


1. VOCABULARY What is the difference between similar fi gures and congruent fi gures?

2. COMPLETE THE SENTENCE A transformation that produces a similar fi gure, such as a dilation,

is called a _________.


In Exercises 3–6, graph △FGH with vertices F(−2, 2),G(−2, −4), and H(−4, −4) and its image after the similarity transformation. (See Example 1.)



4. Dilation: (x, y) → ( 1 — 2 x,

1 —

2 y )




6. Dilation: (x, y) → ( 3 — 4 x,

3 —

4 y )


In Exercises 7 and 8, describe a similarity transformation that maps the blue preimage to the green image. (See Example 2.)

7.

x

y

2

−4

−4−6

F

ED

V

T U

8. L

K

JM

QR

SPx

y

6

4 62−2

In Exercises 9–12, determine whether the polygons with the given vertices are similar. Use transformations to explain your reasoning.

9. A(6, 0), B(9, 6), C(12, 6) and D(0, 3), E(1, 5), F(2, 5)

10. Q(−1, 0), R(−2, 2), S(1, 3), T(2, 1) and

W(0, 2), X(4, 4), Y(6, −2), Z(2, −4)

11. G(−2, 3), H(4, 3), I(4, 0) and

J(1, 0), K(6, −2), L(1, −2)

12. D(−4, 3), E(−2, 3), F(−1, 1), G(−4, 1) and

L(1, −1), M(3, −1), N(6, −3), P(1, −3)

In Exercises 13 and 14, prove that the fi gures are similar.

13. Given Right isosceles △ABC with leg length j,right isosceles △RST with leg length k,

— CA � — RT

Prove △ABC is similar to △RST.

R

S

AC

B

T

jk

14. Given Rectangle JKLM with side lengths x and y,

rectangle QRST with side lengths 2x and 2y

Prove Rectangle JKLM is similar to rectangle QRST.

J K

LM y

x

T

R

S

Q

2y

2x



HSCC_GEOM_PE_04.06.indd 219 5/28/14 3:58 PM 7. Sample answer: translation 1 unit down

and 1 unit right followed by a dilation with

center at E(2, –3) and a scale factor of 2

8. Sample answer: dilation with center at the

origin and a scale factor of 1 —

2 followed by a

refl ection in the y-axis

9. yes; △ABC can be mapped to △DEF by a

dilation with center at the origin and a scale

factor of 1 —

3 followed by a translation of 2

units left and 3 units up.

10. yes; ▭QRST can be mapped to ▭WXYZ

by a 270° rotation about the origin followed

by a dilation with center at the origin and a

scale factor of 2.

11. no; The scale factor from — HI to

— JL is 2 —

3 , but

the scale factor from — GH to

— KL is 5 —

6 .

12. no; The scale factor from — DG to

— LP is 1, but

the scale factor from — FG to

— NP is 5 —

3 .



ASSIGNMENT

Basic: 1, 2, 3–15 odd, 16–18, 23–26

Average: 1, 2, 4–14 even, 16–18, 21–26

Advanced: 1, 2, 8, 12, 14–18, 20–26

HOMEWORK CHECK

Basic: 5, 7, 9, 18

Average: 8, 10, 14, 18

Advanced: 8, 14, 18, 20

ANSWERS 1. Congruent fi gures have the same size

and shape. Similar fi gures have the

same shape, but not necessarily the

same size.

2. similarity transformation

3.

4.

5.

6.

x

4

2

6

−4

y

−4G

G′

F

H

G″

F ″

H″

F′

H′

x

2

−4 2

y

−4G

G′

F

H

G″

F ″

H″

F′

H′

x4 8 12−4

y

−12

GG′

F

H G″F ″

H″

F′

H′

x

2

4

−4

y

−2

−4G

G′

F

H

G″

F ″

H″

F′

H′










220 Chapter 4


15. MODELING WITH MATHEMATICS Determine whether

the regular-sized stop sign and the stop sign sticker

are similar. Use transformations to explain

your reasoning.

12.6 in.

4 in.


comparing the fi gures.

x

y

4

6

2

42 86 1210 14

B

A

Figure A is similar to Figure B.

✗

17. MAKING AN ARGUMENT A member of the

homecoming decorating committee gives a printing

company a banner that is 3 inches by 14 inches to

enlarge. The committee member claims the banner

she receives is distorted. Do you think the printing

company distorted the image she gave it? Explain.

84 in.

18 in.

18. HOW DO YOU SEE IT? Determine whether each pair

of fi gures is similar. Explain your reasoning.

a. b.

19. ANALYZING RELATIONSHIPS Graph a polygon in

a coordinate plane. Use a similarity transformation

involving a dilation (where k is a whole number)

and a translation to graph a second polygon. Then

describe a similarity transformation that maps the

second polygon onto the fi rst.

20. THOUGHT PROVOKING Is the composition of a

rotation and a dilation commutative? (In other words,

do you obtain the same image regardless of the order

in which you perform the transformations?) Justify

your answer.

21. MATHEMATICAL CONNECTIONS Quadrilateral

JKLM is mapped to quadrilateral J′K′L′M′ using

the dilation (x, y) → ( 3 — 2 x,

3 —

2 y ) . Then quadrilateral

J′K′L′M′ is mapped to quadrilateral J″K″L″M″ using

the translation (x, y) → (x + 3, y − 4). The vertices

of quadrilateral J′K′L′M′ are J(−12, 0), K(−12, 18),

L(−6, 18), and M(−6, 0). Find the coordinates of

the vertices of quadrilateral JKLM and quadrilateral

J″K″L″M″. Are quadrilateral JKLM and quadrilateral

J″K″L″M″ similar? Explain.

22. REPEATED REASONING Use the diagram.

x

y

4

6

2

42 6

R

Q S

a. Connect the midpoints of the sides of △QRS

to make another triangle. Is this triangle similar

to △QRS? Use transformations to support

your answer.

b. Repeat part (a) for two other triangles. What

conjecture can you make?

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyClassify the angle as acute, obtuse, right, or straight. (Section 1.5)

23.

113°

24. 25.

82°

26.



ANSWERS 15. yes; The stop sign sticker can be

mapped to the regular-sized stop

sign by translating the sticker to the

left until the centers match, and then

dilating the sticker with a scale factor

of 3.15. Because there is a similarity

transformation that maps one stop

sign to the other, the sticker is similar

to the regular-sized stop sign.


17. no; The scale factor is 6 for both

dimensions. So, the enlarged banner

is proportional to the smaller one.


Mini-Assessment

1. — AB has endpoints A(−8, 6) and B(6, 0). Find the endpoints of its image after the similarity transformation.

Translation: (x, y) → (x, y − 6)

Dilation: (x, y) → ( 1 — 2 x, 1 — 2 y )

x

y8

4

−8

−4

84−8

A′(−8, 0)

B′(6, −6)A″(−4, 0)

B″(3, −3)

A(−8, 6)

B(6, 0)

2. Describe a similarity transformation that maps quadrilateral STUV to quadrilateral DEFG.

x

y4

2

−4

−4

DE

F

G

U

TS

V

Sample answer: A reflection in the x-axis followed by a dilation with a scale factor of 2
















Chapter 4 221

221

4.4–4.6 What Did You Learn?

221

Core VocabularyCore Vocabularycongruent fi gures, p. 200congruence transformation, p. 201dilation, p. 208center of dilation, p. 208scale factor, p. 208

enlargement, p. 208reduction, p. 208similarity transformation, p. 216similar fi gures, p. 216

Core ConceptsCore ConceptsSection 4.4Identifying Congruent Figures, p. 200Describing a Congruence Transformation, p. 201Theorem 4.2 Refl ections in Parallel Lines Theorem, p. 202Theorem 4.3 Refl ections in Intersecting Lines Theorem, p. 203

Section 4.5Dilations and Scale Factor, p. 208Coordinate Rule for Dilations, p. 209

Negative Scale Factors, p. 210

Section 4.6Similarity Transformations, p. 216

Mathematical PracticesMathematical Practices1. Revisit Exercise 33 on page 206. Try to recall the process you used to reach the solution. Did you

have to change course at all? If so, how did you approach the situation?

2. Describe a real-life situation that can be modeled by Exercise 28 on page 213.

Performance Task

The Magic of OpticsLook at yourself in a shiny spoon. What happened to your refl ection? Can you describe this mathematically? Now turn the spoon over and look at your refl ection on the back of the spoon. What happened? Why?

To explore the answers to these questions and more, go to BigIdeasMath.com.

HSCC_GEOM_PE_04.EOC.indd 221 2/24/14 10:40 AM

ANSWERS1. Sample answer: Draw a picture and

label the given information. Then

look at the results and try to fi gure

out what needs to be proven in order

to get there; yes; When unsure, look

back at related defi nitions, postulates,

and theorems to see which ones

might be helpful. Points L and M

must be identifi ed, so use the Ruler

Postulate (Post. 1.1) and Segment

Addition Postulate (Post. 1.2). Then

the rest will start falling into place.

2. Sample answer: This drawing could

represent the reduction of a 16 × 28

painting into a 4 × 7 photograph or

computer graphic.

HSCC_GEOM_TE_04EC.indd 221HSCC_GEOM_TE_04EC.indd 221 6/5/14 1:52 PM6/5/14 1:52 PM

222 Chapter 4


44 Chapter Review

Translations (pp. 173–180)4.1

Graph quadrilateral ABCD with vertices A(1, −2), B(3, −1), C(0, 3), and D(−4, 1) and its image after the translation (x, y) → (x + 2, y − 2).

Graph quadrilateral ABCD. To fi nd the coordinates of the vertices of the

image, add 2 to the x-coordinates and subtract 2 from the y-coordinates

of the vertices of the preimage. Then graph the image.

(x, y) → (x + 2, y − 2)

A(1, −2) → A′(3, −4)

B(3, −1) → B′(5, −3)

C(0, 3) → C′(2, 1)

D(−4, 1) → D′(−2, −1)

Graph △XYZ with vertices X(2, 3), Y(−3, 2), and Z(−4, −3) and its image after the translation.

1. (x, y) → (x, y + 2) 2. (x, y) → (x − 3, y)

3. (x, y) → (x + 3, y − 1) 4. (x, y) → (x + 4, y + 1)

Graph △PQR with vertices P(0, −4), Q(1, 3), and R(2, −5) and its image after the composition.

5. Translation: (x, y) → (x + 1, y + 2) 6. Translation: (x, y) → (x, y + 3)

Translation: (x, y) → (x − 4, y + 1) Translation: (x, y) → (x − 1, y + 1)

Refl ections (pp. 181–188)4.2

Graph △ABC with vertices A(1, −1), B(3, 2), and C(4, −4) and its image in the line y = x.

Graph △ABC. Then use the coordinate rule for refl ecting in the

line y = x to fi nd the coordinates of the endpoints of the image.

(a, b) → (b, a)

A(1, −1) → A′(−1, 1)

B(3, 2) → B′(2, 3)

C(4, −4) → C′(−4, 4)

Graph the polygon and its image after a refl ection in the given line.

7. x = 4

AC

B

x

y

4

2

4 62

8. y = 3

E

GH

F

x

y4

2

4 6

9. How many lines of symmetry does the fi gure have?

x

y4

−4

4−4

D

A

C

B′A′

C′

D′ B

x

y4

−4

−2

4−2−4A

C

B′

A′

C′

B

y = x


HSCC_GEOM_PE_04.EOC.indd 222 5/28/14 3:59 PM

ANSWERS 1.

2.

3.

4.

5.

6.

x

4

2

2−2

y

−2

X

Z′

Y

Z

Y′X′

x

2

2−2−4−6

y

−2

X

Z′

Y

Z

Y′X′

x

2

2 4−2−4

y

−4

X

Z′

Y

Z

Y′

X′

x

2

4

2 4 6−2−4

y

−2

X

Z′

Y

Z

Y′X′

x

4

6

2

2−2

y

−4 PP′

Q′

R′P″

Q″

R″

Q

R

x

6

2

2

y

−2

−4 P

P′

Q′

R′P″

Q″

R″

Q

R

7.

x

4

2

62

y

A A′

B′

C′

B

C

x = 4 8.

9. 2

x

4

6

2

62 4

y

FEF′E′

H′ G′

GH

y = 3


Chapter 4 223

Chapter 4 Chapter Review 223

Rotations (pp. 189–196)4.3

Graph △LMN with vertices L(1, −1), M(2, 3), and N(4, 0) and its image after a 270° rotation about the origin.

Use the coordinate rule for a 270° rotation to fi nd the coordinates

of the vertices of the image. Then graph △LMN and its image.

(a, b) → (b, −a)

L(1, −1) → L′(−1, −1)

M(2, 3) → M′(3, −2)

N(4, 0) → N′(0, −4)

Graph the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin.

10. A(−3, −1), B(2, 2), C(3, −3); 90° 11. W(−2, −1), X(−1, 3), Y(3, 3), Z(3, −3); 180°

12. Graph — XY with endpoints X(5, −2) and Y(3, −3) and its image after a refl ection in

the x-axis and then a rotation of 270° about the origin.

Determine whether the fi gure has rotational symmetry. If so, describe any rotations that map the fi gure onto itself.

13. 14.

Congruence and Transformations (pp. 199–206)4.4

Describe a congruence transformation that maps quadrilateral ABCD to quadrilateral WXYZ, as shown at the right. — AB falls from left to right, and

— WX rises from left to right. If you refl ect

quadrilateral ABCD in the x-axis as shown at the bottom right, then the

image, quadrilateral A′B′C′D′, will have the same orientation as

quadrilateral WXYZ. Then you can map quadrilateral A′B′C′D′ to

quadrilateral WXYZ using a translation of 5 units left.

So, a congruence transformation that maps quadrilateral ABCD to

quadrilateral WXYZ is a refl ection in the x-axis followed by a

translation of 5 units left.

Describe a congruence transformation that maps △DEF to △JKL.

15. D(2, −1), E(4, 1), F(1, 2) and J(−2, −4), K(−4, −2), L(−1, −1)

16. D(−3, −4), E(−5, −1), F(−1, 1) and J(1, 4), K(−1, 1), L(3, −1)

17. Which transformation is the same as refl ecting an object in two

parallel lines? in two intersecting lines?

x

y4

2

4−2−4

M

NL

M′

N′

L′

A

B

C

D

W

Z

YX x

y4

2

42

A

BC

D

W

Z

YX x

y4

2

C′

D′B′

A′


ANSWERS10.

11.

12.

13. yes; Rotations of 60°, 120°, and 180° about the center map the fi gure onto

itself.




y-axis followed by a translation

3 units down


the origin followed by a refl ection in

the line x = 2

17. translation; rotation

x

1

3

1 3−3 −1

y

−3

A

A′

B′C′

B

C

x2

y

−2W

XZ′ Y

ZY′

W′

X′

x

2

2 4

y

−2

−4

XY

Y′X′

Y″

X″


224 Chapter 4


Dilations (pp. 207–214)4.5

Graph trapezoid ABCD with vertices A(1, 1), B(1, 3), C(3, 2), and D(3, 1) and its image after a dilation with a scale factor of 2.

Use the coordinate rule for a dilation with k = 2 to fi nd the coordinates

of the vertices of the image. Then graph trapezoid ABCD and its image.

(x, y) → (2x, 2y)

A(1, 1) → A′(2, 2)

B(1, 3) → B′(2, 6)

C(3, 2) → C′(6, 4)

D(3, 1) → D′(6, 2)

Graph the triangle and its image after a dilation with scale factor k.

18. P(2, 2), Q(4, 4), R(8, 2); k = 1 —

2

19. X(−3, 2), Y(2, 3), Z(1, −1); k = −3

20. You are using a magnifying glass that shows the image of an object that is eight times the

object’s actual size. The image length is 15.2 centimeters. Find the actual length of the object.

Similarity and Transformations (pp. 215–220)4.6

Describe a similarity transformation that maps △FGH to △LMN, as shown at the right. — FG is horizontal, and

— LM is vertical. If you rotate △FGH 90° about the origin as shown at the bottom right, then the image,

△F′G′H′, will have the same orientation as △LMN. △LMN appears to be half as large as △F′G′H′. Dilate △F′G′H′ using a scale factor of

1 —

2 .

(x, y) → ( 1 — 2 x, 1 — 2 y ) F′(−2, 2) → F ″(−1, 1)

G′(−2, 6) → G ″(−1, 3)

H′(−6, 4) → H ″(−3, 2)

The vertices of △F ″G ″H ″ match the vertices of △LMN.

So, a similarity transformation that maps △FGH to △LMN

is a rotation of 90° about the origin followed by a dilation

with a scale factor of 1 —

2 .

Describe a similarity transformation that maps △ABC to △RST.

21. A(1, 0), B(−2, −1), C(−1, −2) and R(−3, 0), S(6, −3), T(3, −6)

22. A(6, 4), B(−2, 0), C(−4, 2) and R(2, 3), S(0, −1), T(1, −2)

23. A(3, −2), B(0, 4), C(−1, −3) and R(−4, −6), S(8, 0), T(−6, 2)

x

y

4

2

6

42 6A

B

D

CD′

C′

B′

A′

H

GFN

L

M

x

y6

2

−2

4 62−2−4−6

H

GFNL

M

G′

H′

F′x

y6

2

−2

4 62−2−4−6


ANSWERS 18.

19.

20. 1.9 cm

21. Sample answer: refl ection in the line

x = −1 followed by a dilation with

center (−3, 0) and k = 3

22. Sample answer: dilation with center

at the origin and k = 1 —

2 , followed by a

refl ection in the line y = x


the origin followed by a dilation with

center at the origin and k = 2

x

4

2

2 4 6 8

y

P

P′

Q′

R′

Q

R

x

4

4 8

y

−4

−10

−6

XZ′ Y

Z

Y′

X′


Chapter 4 225

Chapter 4 Chapter Test 225

Chapter Test44Graph △RST with vertices R(−4, 1), S(−2, 2), and T(3, −2) and its image after the translation.

1. (x, y) → (x − 4, y + 1) 2. (x, y) → (x + 2, y − 2)

Graph the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin.

3. D(−1, −1), E(−3, 2), F(1, 4); 270° 4. J(−1, 1), K(3, 3), L(4, −3), M(0, −2); 90°

Determine whether the polygons with the given vertices are congruent or similar. Use transformations to explain your reasoning.

5. Q(2, 4), R(5, 4), S(6, 2), T(1, 2) and 6. A(−6, 6), B(−6, 2), C(−2, −4) and

W(6, −12), X(15, −12), Y(18, −6), Z(3, −6) D(9, 7), E(5, 7), F(−1, 3)

Determine whether the object has line symmetry and whether it has rotational symmetry. Identify all lines of symmetry and angles of rotation that map the fi gure onto itself.

7. 8. 9.

10. Draw a diagram using a coordinate plane, two parallel lines, and a parallelogram that

demonstrates the Refl ections in Parallel Lines Theorem (Theorem 4.2).

11. A rectangle with vertices W(−2, 4), X(2, 4), Y(2, 2), and Z(−2, 2) is refl ected in the

y-axis. Your friend says that the image, rectangle W′X′Y′Z′, is exactly the same as the

preimage. Is your friend correct? Explain your reasoning.

12. Write a composition of transformations that maps △ABC onto △CDB in the tesselation

shown. Is the composition a congruence transformation? Explain your reasoning.

13. There is one slice of a large pizza and one slice of a

small pizza in the box.

a. Describe a similarity transformation that maps pizza slice

ABC to pizza slice DEF.

b. What is one possible scale factor for a medium slice of pizza?

Explain your reasoning. (Use a dilation on the large slice of pizza.)

14. The original photograph shown is 4 inches by 6 inches.

a. What transfomations can you use to produce the new photograph?

b. You dilate the original photograph by a scale factor of 1 —

2 . What

are the dimensions of the new photograph?

c. You have a frame that holds photos that are 8.5 inches by

11 inches. Can you dilate the original photograph to fi t the

frame? Explain your reasoning.

x

y4

2

−2

42−2−4 A

B

C

E

F

D

new

original

x

y

4

2

0420 86

A B

DC


ANSWERS1.

2.

3.

4.

5. similar; Quadrilateral QRST can be

mapped to quadrilateral WXYZ by a

dilation with the center at the origin

and k = 3, followed by a refl ection in

the x-axis. Because this composition

has a rigid motion and a dilation, it is

a similarity transformation.

6. congruent; △ABC can be mapped to

△DEF by a 270° rotation about the

origin followed by a translation 1 unit

up and 3 units right. Because this is a

composition of two rigid motions, the

composition is rigid.

7. yes, yes; The lines of symmetry are

vertically through the center of the

ball and horizontally through the

center of the ball; 180° 8. yes, no; The line of symmetry runs

from the center of the base of the

guitar, and through the sound hole

to the center of the headstock of the

guitar.

9. no, yes; 180° 10–14. See Additional Answers.

x

2

2−4−6−8

y

−2

SS′

R′

T′T

R

x

2

2 4−4 −2

y

−2

−4

S

S′

R′

T′

T

R

x

4

2

2−2

y

D

D′

E′

F′

E

F

x

4

2

2 4−2

y

−2J′

K′L′

M′

L

M

J

K


Lesson Tutorials Resources by Chapter• Enrichment and Extension• Cumulative Review

Skills Review Handbook Performance Task

BigIdeasMath.com Start the next Section


226 Chapter 4


44 Standards Assessment

1. Which composition of transformations maps △ABC to △DEF? (HSG-CO.A.5)

A

F

D

C

B

x

y4

−4

42−2−4 E

○A Rotation: 90° counterclockwise about the origin


○B Translation: (x, y) → (x − 4, y − 3)

Rotation: 90° counterclockwise about the origin

○C Translation: (x, y) → (x + 4, y − 3)

Rotation: 90° counterclockwise about the origin

○D Rotation: 90° counterclockwise about the origin


2. Use the diagrams to describe the steps you would take to construct a line perpendicular

to line m through point P, which is not on line m. (HSG-CO.D.12)

Step 1 Step 2 Step 3

A Bm

PP

A Bm

P

B

P

A Bm

P

Q

3. Your friend claims that she can fi nd the perimeter of the school crossing

sign without using the Distance Formula. Do you support your friend’s claim?

Explain your reasoning. (HSG-GPE.B.7)

x

y4

−2

4−2


ANSWERS 1. B

2. Step 1. Place the compass at P. Draw

an arc that intersects line m in two

different places. Label the points of

intersection A and B.

Step 2. With the compass at A, draw

an arc below line m using a setting

greater than 1 —

2 AB. Using the same

compass setting, draw an arc from B

that intersects the previous arc. Label

the intersection Q.

Step 3. Use a straightedge to draw — PQ .

3. yes; She could fi nd the side lengths

and the bottom length by counting

units, and then fi nd the angled lengths

using the Pythagorean Theorem.


Chapter 4 227

Chapter 4 Standards Assessment 227

4. Graph the directed line segment ST with endpoints S(−3, −2) and T(4, 5). Then fi nd

the coordinates of point P along the directed line segment ST so that the ratio of SP to

PT is 3 to 4. (HSG-GPE.B.6)

5. The graph shows quadrilateral WXYZ and quadrilateral ABCD. (HSG-CO.B.6)

A

B

C

D

W

Z

Y

X

x

y

−4

−2

42−4

a. Write a composition of transformations that maps quadrilateral WXYZ to

quadrilateral ABCD.

b. Are the quadrilaterals congruent? Explain your reasoning.

6. Which equation represents the line passing through the point (−6, 3) that is parallel to

the line y = − 1 —

3 x − 5? (HSG-GPE.B.5)

○A y = 3x + 21

○B y = − 1 — 3 x − 5

○C y = 3x − 15

○D y = − 1 — 3 x + 1

7. Which scale factor(s) would create a dilation of — AB that is shorter than

— AB ? Select all

that apply. (HSG-SRT.A.1b)

3

— 4

2 7 —

2

1 —

2 31

1 —

3

3 —

2

A B

8. List one possible set of coordinates of the vertices of quadrilateral ABCD for each

description. (HSG-CO.A.3)

a. A refl ection in the y-axis maps quadrilateral ABCD onto itself.

b. A refl ection in the x-axis maps quadrilateral ABCD onto itself.

c. A rotation of 90° about the origin maps quadrilateral ABCD onto itself.

d. A rotation of 180° about the origin maps quadrilateral ABCD onto itself.


ANSWERS4. (0, 1)

5. a. Sample answer: 90° rotation

about the origin followed by a

refl ection in the x-axis

b. yes; The transformations used

to map quadrilateral WXYZ

to quadrilateral ABCD are

rigid, so this is a congruence

transformation.

6. D

7. 1 —

3 ,

1 —

2 ,

3 —

4

8. a. Sample answer: A(2, 0), B(2, 5),

C(−2, 5), D(−2, 0)

b. Sample answer: A(0, 2), B(3, 2),

C(3, −2), D(0, −2)

c. Sample answer: A(2, 0), B(0, 2),

C(−2, 0), D(0, −2)

d. Sample answer: A(2, 2),

B(−1, 1), C(−2, −2), D(1, −1)

x

2

4

2 4−2

y

−2

T

P(0, 1)

S


4 Transformations - Mathematics -...

Documents

Transcript of 4 Transformations - Mathematics -...