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11. The figure shows the ascent of a hot air balloon. Howwould you describe the translation?4212. Critical Thinking Is it possible that the orientation of afigure could change after it is translated? Explain.-4-242-4FOCUS ON HIGHER ORDER THINKING13. a. Multistep Graph triangle XYZ with vertices X(–2, –5),Y(2, –2), and Z(4, –4) on the coordinate grid.y6b. On the same coordinate grid, graph and labeltriangle X ′Y ′Z ′, the image of triangle XYZ after atranslation of 3 units to the left and 6 units up.x-6d. Analyze Relationships How would you describethe translation that maps triangle XYZ onto triangleX ′′Y ′′Z ′′?y6-6O-6286Unit 46-614. Critical Thinking The figure shows rectangle P ′Q ′R ′S ′,the image of rectangle PQRS after a translation of5 units to the right and 7 units up. Graph and labelthe preimage PQRS.15. Communicate Mathematical Ideas Explain why theimage of a figure after a translation is congruent to itspreimage.OP′Q′S′R′x6 Houghton Mifflin Harcourt Publishing Companyc. Now graph and label triangle X ′′Y ′′Z ′′, the image oftriangle X ′Y ′Z ′ after a translation of 1 unit to the leftand 2 units down.

LESSON9.2?Properties ofReflections8.G.1bVerify experimentally theproperties of rotations,reflections, and translations:Angles are taken to anglesof the same measure. Also8.G.1a, 8.G.1c, 8.G.3ESSENTIAL QUESTIONHow do you describe the properties of reflection and their effecton the congruence and orientation of figures?EXPLORE ACTIVITY 18.G.1bExploring ReflectionsA reflection is a transformation that flips a figure across a line.The line is called the line of reflection. Each point and its imageare the same distance from the line of reflection.yThe triangle shown on the grid is the preimage. You willexplore reflections across the x- and y-axes.6A Trace triangle ABC and the x- and y-axes onto a piece ofpaper.B Fold your paper along the x-axis and trace the image ofthe triangle on the opposite side of the x-axis. Unfold yourpaper and label the vertices of the image A′, B ′, and C ′.ACBxO-66ª )PVHIUPO .JGGMJO )BSDPVSU 1VCMJTIJOH PNQBOZ t *NBHF SFEJUT ª(FUUZ *NBHFT 1IPUPEJTDC What is the line of reflection for this transformation?-6D Find the perpendicular distance from each point to theline of reflection.Point APoint BPoint CE Find the perpendicular distance from each point to the line of reflection.Point A'Point B 'Point C 'F What do you notice about the distances you found inDandE?Reflect1. Fold your paper from A along the y-axis and trace the image oftriangle ABC on the opposite side. Label the vertices of the imageA″, B″, and C″. What is the line of reflection for this transformation?2. How does each image in your drawings compare with its preimage?Lesson 9.2287

EXPLORE ACTIVITY 28.G.1bProperties of ReflectionsyUse trapezoid TRAP to investigate the propertiesof reflections.TRA Trace the trapezoid onto a piece of paper. Cut outyour traced trapezoid.B Place your trapezoid on top of the trapezoid in thefigure. Then reflect your trapezoid across the y-axis.Sketch the image of the reflection by tracing yourtrapezoid in this new location. Label the vertices ofthe image T ′, R ′, A′, and P ′.6AxPO-66-4C Use a ruler to measure the sides of trapezoid TRAP in centimeters.TR RA AP TP D Use a ruler to measure the sides of trapezoid T ′R ′A′P ′ in centimeters.T ′R ′ R ′A′ A′P ′ T ′P ′ E What do you notice about the lengths of corresponding sides ofthe two figures?F Use a protractor to measure the angles of trapezoid TRAP.m T m R m A m P m T ′ m R ′ m A′ m P ′ H What do you notice about the measures of corresponding angles of thetwo figures?I Which sides of trapezoid TRAP are parallel?Which sides of trapezoid T ′R ′A′P ′ are parallel?What do you notice?288Unit 4 Houghton Mifflin Harcourt Publishing CompanyG Use a protractor to measure the angles of trapezoid T ′R ′A′P ′.

Reflect3. Make a Conjecture Use your results fromconjecture about reflections.E,H, andIto make aMath TalkMathematical PracticesWhat can yousay about reflectionsand congruence?Graphing ReflectionsTo reflect a figure across a line of reflection, reflect each of its vertices. Thenconnect the vertices to form the image. Remember that each point and itsimage are the same distance from the line of reflection.Math On the Spotmy.hrw.comEXAMPL 1EXAMPLE8.G.3The figure shows triangle XYZ. Graph the image of the triangle after areflection across the x-axis.STEP 1yReflect point X.6Point X is 3 units below thex-axis. Count 3 units abovethe x-axis and plot point X′ .STEP 2Reflect point Y.Point Y is 1 unit below thex-axis. Count 1 unit abovethe x-axis and plot point Y ′. Houghton Mifflin Harcourt Publishing CompanySTEP 3X'3-6OXPoint Z is 5 units below thex-axis. Count 5 units abovethe x-axis and plot point Z ′.STEP 4x5ZyConnect X ′, Y ′, and Z ′ to formtriangle X ′Y ′Z ′.Each vertex of the imageis the same distancefrom the x-axis as thecorresponding vertex inthe original figure.35Y'1Y 1-6Reflect point Z.My NotesZ'6Z'X'-6xY'OY6X-6ZLesson 9.2289

yYOUR TURNPersonalMath TrainerOnline Practiceand HelpBA4. The figure shows pentagonABCDE. Graph the image of thepentagon after a reflection acrossthe y-axis.6CxDE-6my.hrw.comO6-6Guided Practice1. Vocabulary A reflection is a transformation that flips a figure acrossa line called the.2. The figure shows trapezoid ABCD. (Explore Activities 1 and 2 and Example 1)ya. Graph the image of the trapezoid after a reflectionacross the x-axis. Label the vertices of the image.6BDc. What If? Suppose you reflected trapezoid ABCDacross the y-axis. How would the orientation ofthe image of the trapezoid compare with theorientation of the preimage?-6-6?ESSENTIAL QUESTION CHECK-IN3. What are the properties of reflections?290Unit 4xCO6 Houghton Mifflin Harcourt Publishing CompanyAb. How do trapezoid ABCD and trapezoid A'B'C'D'compare?

NameClassDate9.2 Independent PracticePersonalMath Trainer8.G.1a, 8.G.1b, 8.G.1c, 8.G.3my.hrw.comThe graph shows four right triangles. Use thegraph for Exercises 4–7.y8. a. Graph quadrilateral WXYZ withvertices W(-2, -2), X(3, 1), Y(5, -1),and Z(4, -6) on the coordinate grid.y6AOnline Practiceand HelpB462x-6-4-2O24-2C-4X6-6DxOY6W-6-64. Which two triangles are reflections of eachother across the x-axis?5. For which two triangles is the line ofreflection the y-axis? Houghton Mifflin Harcourt Publishing Company6. Which triangle is a translation of triangle C?How would you describe the translation?7. Which triangles are congruent? How doyou know?Zb. On the same coordinate grid, graphquadrilateral W′X′Y′Z′, the image ofquadrilateral WXYZ after a reflectionacross the x-axis.c. Which sideof the image is congruentto side YZ?Name three other pairs of congruentsides.d. Which angle of the image is congruentto X?Name three other pairs of congruentangles.Lesson 9.2291

9. Critical Thinking Is it possible that the image of a point after a reflectioncould be the same point as the preimage? Explain.FOCUS ON HIGHER ORDER THINKINGy10. a. Graph the image of the figure shown after a reflectionacross the y-axis.5b. On the same coordinate grid, graph the image of the figureyou drew in part a after a reflection across the x-axis.xO-5c. Make a Conjecture What other sequence oftransformations would produce the same final imagefrom the original preimage? Check your answer byperforming the transformations. Then make a conjecturethat generalizes your findings.-5y11. a. Graph triangle DEF with vertices D(2, 6), E(5, 6), andF(5, 1) on the coordinate grid.6b. Next graph triangle D ′E ′F ′, the image of triangleDEF after a reflection across the y-axis.d. Analyze Relationships Find a different sequenceof transformations that will transform triangle DEFto triangle D ′′E ′′F ′′.292Unit 4DE42xF-6-4-2O-2-4-6246 Houghton Mifflin Harcourt Publishing Companyc. On the same coordinate grid, graph triangle D′′ E′′ F′′,the image of triangle D′E′F′ after a translation of7 units down and 2 units to the right.5

LESSON9.3?Properties ofRotationsESSENTIAL QUESTION8.G.1cVerify experimentally theproperties of rotations,reflections, and translations:Parallel lines are taken toparallel lines. Also 8.G.1a,8.G.1b, 8.G.3How do you describe the properties of rotation and their effect onthe congruence and orientation of figures?EXPLORE ACTIVITY 18.G.1cExploring RotationsA rotation is a transformation that turns a figure around a givenpoint called the center of rotation. The image has the same sizeand shape as the preimage.The triangle shown on the grid is the preimage. You will usethe origin as the center of rotation.A Trace triangle ABC onto a piece of paper. Cut out yourtraced triangle.y5B Rotate your triangle 90 counterclockwise about theorigin. The side of the triangle that lies along the x-axisshould now lie along the y-axis.ª )PVHIUPO .JGGMJO )BSDPVSU 1VCMJTIJOH PNQBOZ t *NBHF SFEJUT ª*,0 'PUPMJBC Sketch the image of the rotation. Label the images ofpoints A, B, and C as A′, B′, and C′.CAxB-55D Describe the motion modeled by the rotation.Rotateabout the origin.degrees-5E Check that the motion you described in D is the same motion thatmaps point A onto A′, point B onto B′, and point C onto C′.Reflect1.Communicate Mathematical Ideas How are the size and theorientation of the triangle affected by the rotation?2.Rotate triangle ABC 90 clockwise about the origin. Sketch the resulton the coordinate grid above. Label the image vertices A″, B″, and C″.Lesson 9.3293

EXPLORE ACTIVITY 28.G.1cProperties of RotationsUse trapezoid TRAP to investigate the properties of rotations.yA Trace the trapezoid onto a piece of paper.Include the portion of the x- and y-axesbordering the third quadrant. Cut out yourtracing.B Place your trapezoid and axes on top of thosein the figure. Then use the axes to help rotateyour trapezoid 180 counterclockwise aboutthe origin. Sketch the image of the rotation ofyour trapezoid in this new location. Label thevertices of the image T′, R′, A′, and P′.6C Use a ruler to measure the sides of trapezoidTRAP in centimeters.TR RA AP TP xO-6R-6T6APCut along lineafter tracing.D Use a ruler to measure the sides of trapezoid T′R′A′P′ in centimeters.T′R′ R′A′ A′P′ T′P′ E What do you notice about the lengths of corresponding sides of thetwo figures?F Use a protractor to measure the angles of trapezoid TRAP.m R m A m P G Use a protractor to measure the angles of trapezoid T′R′A′P′.m T′ m R′ m A′ m P′ H What do you notice about the measures of corresponding angles of thetwo figures?I Which sides of trapezoid TRAP are parallel?Which sides of trapezoid T′R′A′P′ are parallel?What do you notice?294Unit 4 Houghton Mifflin Harcourt Publishing Companym T

Reflect3.Make a Conjecture Use your results fromconjecture about rotations.4.Place your tracing back in its original position. Then perform a 180 clockwise rotation about the origin. Compare the result with the result inE,H, andto make aIB.Graphing RotationsTo rotate a figure in the coordinate plane, rotate each of its vertices. Thenconnect the vertices to form the image.Math On the SpotEXAMPL 1EXAMPLE8.G.3yThe figure shows triangle ABC. Graph theimage of triangle ABC after a rotation of90 clockwise.STEP 1Rotate the figure clockwise fromthe y-axis to the x-axis. Point Awill still be at (0, 0).5B Houghton Mifflin Harcourt Publishing CompanyC2-5-25AnimatedMathmy.hrw.comC′Math TalkMathematical PracticesHow is the orientationof the triangle affectedby the rotation?yConnect A′, B′, and C′ to form theimage triangle A′B′C′.5BReflectIs the image congruent to thepreimage? How do you know?2-5Point C is 2 units to the right ofthe y-axis, so point C′ is 2 unitsbelow the x-axis.5.B′xAA′-2Point B is 2 units to the left ofthe y-axis, so point B′ is 2 unitsabove the x-axis.STEP 2my.hrw.comC2-5-2AA′-2B′x25C′-5Lesson 9.3295

YOUR TURNyGraph the image of quadrilateral ABCDafter each rotation.PersonalMath TrainerOnline Practiceand Help56. 180 my.hrw.comB7. 270 clockwiseCx8. Find the coordinates of Point C after a90 counterclockwise rotation followedby a 180 rotation.A-5D5-5Guided Practice1. Vocabulary A rotation is a transformation that turns a figure around agivencalled the center of rotation.Siobhan rotates a right triangle 90 counterclockwise about the origin.2. How does the orientation of the image of the triangle compare with theorientation of the preimage? (Explore Activity 1)3. Is the image of the triangle congruent to the preimage? (Explore Activity 2)Draw the image of the figure after the given rotation about the origin. (Example 1)5. 180 y44-5?xEO-4F5GESSENTIAL QUESTION CHECK-IN6. What are the properties of rotations?296Unit 4y-5DO-4ABCx5 Houghton Mifflin Harcourt Publishing Company4. 90 counterclockwise

NameClassDate9.3 Independent PracticePersonalMath Trainer8.G.1a, 8.G.1b, 8.G.1c, 8.G.3my.hrw.comOnline Practiceand Helpy7. The figure shows triangle ABC and a rotation of the triangleabout the origin.C'a. How would you describe the rotation?5B'CA'O-55b. What are the coordinates of the image?,BA,x-58. The graph shows a figure and its image after atransformation.2a. How would you describe this as a rotation?-55-2b. Can you describe this as a transformation other than arotation? Explain.y9. What type of rotation will preserve the orientationof the H-shaped figure in the grid?3 Houghton Mifflin Harcourt Publishing Companyx10. A point with coordinates (-2, -3) is rotated 90 clockwiseabout the origin. What are the coordinates of its image?-3O3-3Complete the table with rotations of 180 or 90 . Include thedirection of rotation for rotations of 90 .Shape in quadrantImage in quadrant11.IIV12.IIII13.IVIIIRotationLesson 9.3297

Draw the image of the figure after the given rotation about the origin.14. 180 15. 270 counterclockwiseyy5CB-5xADC5BA-55-5Ox5-516. Is there a rotation for which the orientation of the image is always thesame as that of the preimage? If so, what?Work AreaFOCUS ON HIGHER ORDER THINKING17. Problem Solving Lucas is playing a game where he has to rotate a figurefor it to fit in an open space. Every time he clicks a button, the figurerotates 90 degrees clockwise. How many times does he need to click thebutton so that each figure returns to its original orientation?Figure AABCFigure B18. Make a Conjecture Triangle ABC is reflected across the y-axis to formthe image A′B′C′. Triangle A′B′C′ is then reflected across the x-axis to formthe image A″B″C″. What type of rotation can be used to describe therelationship between triangle A″B″C″ and triangle ABC?19. Communicate Mathematical Ideas Point A is on the y-axis. Describe allpossible locations of image A′ for rotations of 90 , 180 , and 270 . Includethe origin as a possible location for A.298Unit 4 Houghton Mifflin Harcourt Publishing CompanyFigure C

AlgebraicRepresentations ofTransformationsLESSON9.4?ESSENTIAL QUESTION8.G.3Describe the effect ofdilations, translations,rotations, and reflectionson two-dimensional figuresusing coordinates.How can you describe the effect of a translation, rotation, orreflection on coordinates using an algebraic representation?Algebraic Representationsof TranslationsThe rules shown in the table describe how coordinates change when a figure istranslated up, down, right, and left on the coordinate plane.Math On the Spotmy.hrw.comTranslationsRight a unitsAdd a to the x-coordinate: (x, y) (x a, y)Left a unitsSubtract a from the x-coordinate: (x, y) (x - a, y)Up b unitsAdd b to the y-coordinate: (x, y) (x, y b)Down b unitsSubtract b from the y-coordinate: (x, y) (x, y - b)EXAMPL 1EXAMPLE8.G.3 Houghton Mifflin Harcourt Publishing CompanyTriangle XYZ has vertices X(0, 0), Y(2, 3), and Z(4, -1). Find the vertices oftriangle X′Y′Z′ after a translation of 3 units to the right and 1 unit down.Then graph the triangle and its image.Add 3 to the x-coordinate of eachvertex and subtract 1 from they-coordinate of each vertex.STEP 1 Apply the rule to find the vertices of the image.STEP 2Vertices of XYZRule: (x 3, y - 1)Vertices of X′ Y′ Z′X(0, 0)(0 3, 0 - 1)X′(3, -1)Y(2, 3)(2 3, 3 - 1)Y′(5, 2)Z(4, -1)(4 3, -1 - 1)Z′(7, -2)Graph triangle XYZ and its image.y3XO-3YY′xX′Z7Z′Math TalkMathematical PracticesWhen you translate afigure to the left orright, which coordinatedo you change?Lesson 9.4299

YOUR TURNPersonalMath TrainerOnline Practiceand Help1.A rectangle has vertices at (0, -2), (0, 3), (3, -2), and (3, 3). What are thecoordinates of the vertices of the image after the translation(x, y) (x - 6, y - 3)? Describe the translation.my.hrw.comAlgebraic Representationsof ReflectionsMath On the Spotmy.hrw.comThe signs of the coordinates of a figure change when the figure is reflectedacross the x-axis and y-axis. The table shows the rules for changing the signsof the coordinates after a reflection.ReflectionsAcross the x-axisMultiply each y-coordinate by -1: (x, y) (x, -y)Across the y-axisMultiply each x-coordinate by -1: (x, y) (-x, y)EXAMPLE 2Rectangle RSTU has vertices R(-4, -1), S(-1, -1), T(-1, -3), and U(-4, -3).Find the vertices of rectangle R′S′T′U′ after a reflection acrossthe y-axis. Then graph the rectangle and its image.Multiply thex-coordinate ofeach vertex by -1.STEP 1 Apply the rule to find the vertices of the image.Vertices of RSTUR(-4, -1)Vertices of R′S′T′U′(-1 · (-1), - 1)S′(1, -1)(-1 · (-4), - 1)S(-1, -1)T(-1, -3)R′(4, -1)(-1 · (-1), - 3)U(-4, -3)STEP 2Rule: (-1 · x, y)T′(1, -3)(-1 · (-4), - 3)U′(4, -3)Graph rectangle RSTU and its image.y3x-5RSUTO-5300Unit 4S′5R′T′U′ Houghton Mifflin Harcourt Publishing CompanyMy Notes8.G.3

YOUR TURN2. Triangle ABC has vertices A(-2, 6), B(0, 5), and C(3, -1). Find the verticesof triangle A′B′C′ after a reflection across the x-axis.PersonalMath TrainerOnline Practiceand Helpmy.hrw.comAlgebraic Representations of RotationsWhen points are rotated about the origin, the coordinates of the image can befound using the rules shown in the table.Math On the SpotRotations90 clockwiseMultiply each x-coordinate by -1; then switchthe x- and y-coordinates: (x, y) (y, -x)90 counterclockwiseMultiply each y-coordinate by -1; then switchthe x- and y-coordinates: (x, y) (-y, x)180 Multiply both coordinates by -1: (x, y) (-x, -y)EXAMPL 3EXAMPLEmy.hrw.com8.G.3 Houghton Mifflin Harcourt Publishing CompanyQuadrilateral ABCD has vertices at A(-4, 2), B(-3, 4), C(2, 3), and D(0, 0).Find the vertices of quadrilateral A′B′C′D′ after a 90 clockwise rotation.Then graph the quadrilateral and its image.Multiply the x-coordinate ofeach vertex by -1, and thenswitch the x- and y-coordinates.STEP 1 Apply the rule to find the vertices of the image.STEP 2Vertices of ABCDRule: (y, -x)Vertices of A′B′C′D′A(-4, 2)(2, -1 · (-4))A′(2, 4)B(-3, 4)(4, -1 · (-3))B′(4, 3)C(2, 3)(3, -1

Aug 12, 2015 · Personal Math Trainer Interactively explore key concepts to see how math works. Animated Math Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. my.hrw.com Math On the