9 LESSON 9.1. Transformations and Congruence. Properties of Translations ESSENTIAL QUESTION

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1 Transformations and ongruence? MULE 9 LESSN 9.1 ESSENTIL QUESTIN Properties of Translations How can ou use transformations and congruence to solve realworld problems? 8.G.1, 8.G.3 LESSN 9. Properties of Reflections 8.G.1, 8.G.3 LESSN 9.3 Properties of Rotations 8.G.1, 8.G.3 LESSN 9. lgebraic Representations of Transformations Houghton Mifflin Harcourt Publishing ompan Image redits: Gett Images Sport/Gett Images 8.G.3 LESSN 9. ongruent Figures 8.G. Real-World Video m.hrw.com m.hrw.com When a marching band lines up and marches across the field, the are modeling a translation. s the march, the maintain size and orientation. translation is one tpe of transformation. m.hrw.com Math n the Spot nimated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 77

2 re YU Read? omplete these eercises to review skills ou will need for this module. Integer perations m.hrw.com Personal Math Trainer nline Practice and Help EXMPLE -3 - (-) = -3 + = -3 - = 3 To subtract an integer, add its opposite. The signs are different, so find the difference of the absolute values: - 3 = 3. Use the sign of the number with the greater absolute value. Find each difference (-9) (-). 3 - (-11) (-1) Measure ngles EXMPLE J L K m JKL = Place the center point of the protractor on the angle s verte. lign one ra with the base of the protractor. Read the angle measure where the other ra intersects the semicircle. Use a protractor to measure each angle. 9. F 10. X 11. H G Y Z S R T Houghton Mifflin Harcourt Publishing ompan 78 Unit

3 Reading Start-Up Visualize Vocabular Use the words to complete the graphic organizer. You will put one word in each oval. quadrilateral in which all sides are congruent and opposite sides are parallel. Tpes of Quadrilaterals Understand Vocabular quadrilateral in which opposite sides are parallel and congruent. quadrilateral with eactl one pair of parallel sides. Match the term on the left to the correct epression on the right. Vocabular Review Words coordinate plane (plano cartesiano) parallelogram (paralelogramo) quadrilateral (cuadrilátero) rhombus (rombo) trapezoid (trapecio) Preview Words center of rotation (centro de rotación) congruent (congruente) image (imagen) line of reflection (línea de refleión) preimage (imagen original) reflection (refleión) rotation (rotación) transformation (transformación) translation (traslación) 1. transformation. function that describes a change in the position, size, or shape of a figure.. reflection. function that slides a figure along a straight line. Houghton Mifflin Harcourt Publishing ompan 3. translation. transformation that flips a figure across a line. ctive Reading ooklet efore beginning the module, create a booklet to help ou learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. s ou stud each lesson, write important details that support the main idea, such as vocabular and formulas. Refer to our finished booklet as ou work on assignments and stud for tests. Module 9 79

4 GETTING REY FR Transformations and ongruence Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. 8.G. Understand that a twodimensional figure is congruent to another if the second can be obtained from the first b a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that ehibits the congruence between them. What It Means to You You will identif a rotation, a reflection, a translation, and a sequence of transformations, and understand that the image has the same shape and size as the preimage. EXMPLE 8.G. The figure shows triangle and its image after three different transformations. Identif and describe the translation, the reflection, and the rotation of triangle. Figure 1 is a translation units down. Figure is a reflection across the -ais. Figure 3 is a rotation of 180. Figure - Figure 3 - Figure 1 8.G.3 escribe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Visit m.hrw.com to see all ommon ore Standards eplained. m.hrw.com What It Means to You You can use an algebraic representation to translate, reflect, or rotate a two-dimensional figure. EXMPLE 8.G.3 Rectangle RSTU with vertices (-, 1), (-1, 1), (-1, -3), and (-, -3) is reflected across the -ais. Find the coordinates of the image. The rule to reflect across the -ais is to change the sign of the -coordinate. oordinates (-, 1), (-1, 1), (-1, -3), (-, -3) Reflect across the -ais (-, ) (-(-), 1), (-(-1), 1), (- (-1), -3), (-(-), -3) oordinates of image (, 1), (1, 1), (1, -3), (, -3) The coordinates of the image are (, 1), (1, 1), (1, -3), and (, -3). Houghton Mifflin Harcourt Publishing ompan 80 Unit

5 ? LESSN 9.1 Properties of Translations ESSENTIL QUESTIN How do ou describe the properties of translation and their effect on the congruence and orientation of figures? 8.G.1a Verif eperimentall the properties of rotations, reflections, and translations: Lines are taken to lines, and line segments to line segments of the same length. lso 8.G.1b, 8.G.1c, 8.G.3 EXPLRE TIVITY 1 8.G.1a Eploring Translations You learned that a function is a rule that assigns eactl one output to each input. transformation is a function that describes a change in the position, size, or shape of a figure. The input of a transformation is the preimage, and the output of a transformation is the image. translation is a transformation that slides a figure along a straight line. The triangle shown on the grid is the preimage (input). The arrow shows the motion of a translation and how point is translated to point. E F Trace triangle and line onto a piece of paper. Slide our triangle along the line to model the translation that maps point to point. The image of the translation is the triangle produced b the translation. Sketch the image of the translation. The vertices of the image are labeled using prime notation. For eample, the image of is. Label the images of points and. escribe the motion modeled b the translation. Move units right and units down. heck that the motion ou described in part E is the same motion that maps point onto, point onto, and point onto. Reflect 1. How is the orientation of the triangle affected b the translation? Lesson

6 EXPLRE TIVITY 8.G.1a Properties of Translations Use trapezoid TRP to investigate the properties of translations. Trace the trapezoid onto a piece of paper. ut out our traced trapezoid. Place our trapezoid on top of the trapezoid in the figure. Then translate our trapezoid units to the left and 3 units up. Sketch the image of the translation b tracing our trapezoid in this new location. Label the vertices of the image T, R,, and P. - - T R P Use a ruler to measure the sides of trapezoid TRP in centimeters. TR = R = P = TP = Use a ruler to measure the sides of trapezoid T R P in centimeters. T R = R = P = T P = E What do ou notice about the lengths of corresponding sides of the two figures? F Use a protractor to measure the angles of trapezoid TRP. m T = m R = m = m P = G Use a protractor to measure the angles of trapezoid T R P. H I m T = m R = m = m P = What do ou notice about the measures of corresponding angles of the two figures? Which sides of trapezoid TRP are parallel? How do ou know? Which sides of trapezoid T R P are parallel? What do ou notice? Houghton Mifflin Harcourt Publishing ompan 8 Unit

7 Reflect. Make a onjecture Use our results from parts E, H, and I to make a conjecture about translations. 3. Two figures that have the same size and shape are called congruent. What can ou sa about translations and congruence? Graphing Translations To translate a figure in the coordinate plane, translate each of its vertices. Then connect the vertices to form the image. EXMPLE 1 The figure shows triangle XYZ. Graph the image of the triangle after a translation of units to the right and 1 unit up. 8.G.3 Math n the Spot m.hrw.com Houghton Mifflin Harcourt Publishing ompan STEP 1 STEP STEP 3 STEP Translate point X. ount right units and up 1 unit and plot point X. Translate point Y. ount right units and up 1 unit and plot point Y. Translate point Z. ount right units and up 1 unit and plot point Z. onnect X, Y, and Z to form triangle X Y Z. Each verte is moved units right and 1 unit up. X - X - Z Z X Y - X Y - Z Z Y Y Math Talk Mathematical Practices Is the image congruent to the preimage? How do ou know? Lesson

8 Personal Math Trainer nline Practice and Help m.hrw.com YUR TURN. The figure shows parallelogram. Graph the image of the parallelogram after a translation of units to the left and units down. - - Guided Practice 1. Vocabular is a change in the position, size, or shape of a figure.. Vocabular When ou perform a transformation of a figure on the coordinate plane, the input of the transformation is called the, and the output of the transformation is called the. 3. Joni translates a right triangle units down and units to the right. How does the orientation of the image of the triangle compare with the orientation of the preimage? (Eplore ctivit 1). Rashid drew rectangle PQRS on a coordinate plane. He then translated the rectangle 3 units up and 3 units to the left and labeled the image P Q R S. How do rectangle PQRS and rectangle P Q R S compare? (Eplore ctivit )?. The figure shows trapezoid WXYZ. Graph the image of the trapezoid after a translation of units up and units to the left. (Eample 1) ESSENTIL QUESTIN HEK-IN. What are the properties of translations? - W Z - Y X Houghton Mifflin Harcourt Publishing ompan 8 Unit

9 Name lass ate 9.1 Independent Practice 8.G.1a, 8.G.1b, 8.G.1c, 8.G.3 7. The figure shows triangle EF. a. Graph the image of the triangle after the translation that maps point to point. m.hrw.com E Personal Math Trainer nline Practice and Help b. How would ou describe the translation? - F c. How does the image of triangle EF compare with the preimage? - 8. a. Graph quadrilateral KLMN with vertices K( 3, ), L(, ), M(0, 3), and N(, 0) on the coordinate grid. b. n the same coordinate grid, graph the image of quadrilateral KLMN after a translation of 3 units to the right and units up. c. Which side of the image is congruent to side _ LM? - Name three other pairs of congruent sides. - raw the image of the figure after each translation. 9. units left and units down 10. units right and 3 units up Houghton Mifflin Harcourt Publishing ompan - P S Q R Lesson 9.1 8

10 11. The figure shows the ascent of a hot air balloon. How would ou describe the translation? 1. ritical Thinking Is it possible that the orientation of a figure could change after it is translated? Eplain FUS N HIGHER RER THINKING 13. a. Multistep Graph triangle XYZ with vertices X(, ), Y(, ), and Z(, ) on the coordinate grid. b. n the same coordinate grid, graph and label triangle X Y Z, the image of triangle XYZ after a translation of 3 units to the left and units up. c. Now graph and label triangle X Y Z, the image of triangle X Y Z after a translation of 1 unit to the left and units down. - d. nalze Relationships How would ou describe the translation that maps triangle XYZ onto triangle X Y Z? - 1. ritical Thinking The figure shows rectangle P Q R S, the image of rectangle PQRS after a translation of units to the right and 7 units up. Graph and label the preimage PQRS. 1. ommunicate Mathematical Ideas Eplain wh the image of a figure after a translation is congruent to its preimage. - P Q S R Houghton Mifflin Harcourt Publishing ompan - 8 Unit

11 ? LESSN 9. Properties of Reflections ESSENTIL QUESTIN 8.G.1b Verif eperimentall the properties of rotations, reflections, and translations: ngles are taken to angles of the same measure. lso 8.G.1a, 8.G.1c, 8.G.3 How do ou describe the properties of reflection and their effect on the congruence and orientation of figures? EXPLRE TIVITY 1 8.G.1b Eploring Reflections reflection is a transformation that flips a figure across a line. The line is called the line of reflection. Each point and its image are the same distance from the line of reflection. The triangle shown on the grid is the preimage. You will eplore reflections across the - and -aes. Trace triangle and the - and -aes onto a piece of paper. Fold our paper along the -ais and trace the image of the triangle on the opposite side of the -ais. Unfold our paper and label the vertices of the image,, and. - What is the line of reflection for this transformation? E Find the perpendicular distance from each point to the line of reflection. Point Point Point Find the perpendicular distance from each point to the line of reflection. Point ' Point ' Point ' F E What do ou notice about the distances ou found in and? Reflect 1. Fold our paper from along the -ais and trace the image of triangle on the opposite side. Label the vertices of the image,, and. What is the line of reflection for this transformation?. How does each image in our drawings compare with its preimage? - Lesson 9. 87

12 EXPLRE TIVITY 8.G.1b Properties of Reflections Use trapezoid TRP to investigate the properties of reflections. Trace the trapezoid onto a piece of paper. ut out our traced trapezoid. Place our trapezoid on top of the trapezoid in the figure. Then reflect our trapezoid across the -ais. Sketch the image of the reflection b tracing our trapezoid in this new location. Label the vertices of the image T, R,, and P. - T P R - Use a ruler to measure the sides of trapezoid TRP in centimeters. TR = R = P = TP = Use a ruler to measure the sides of trapezoid T R P in centimeters. T R = R = P = T P = E What do ou notice about the lengths of corresponding sides of the two figures? F Use a protractor to measure the angles of trapezoid TRP. m T = m R = m = m P = G H I Use a protractor to measure the angles of trapezoid T R P. m T = m R = m = m P = What do ou notice about the measures of corresponding angles of the two figures? Which sides of trapezoid TRP are parallel? Which sides of trapezoid T R P are parallel? What do ou notice? Houghton Mifflin Harcourt Publishing ompan 88 Unit

13 Reflect 3. Make a onjecture Use our results from E, H, and I to make a conjecture about reflections. Math Talk Mathematical Practices What can ou sa about reflections and congruence? Graphing Reflections To reflect a figure across a line of reflection, reflect each of its vertices. Then connect the vertices to form the image. Remember that each point and its image are the same distance from the line of reflection. EXMPLE 1 8.G.3 Math n the Spot m.hrw.com The figure shows triangle XYZ. Graph the image of the triangle after a reflection across the -ais. STEP 1 STEP STEP 3 Reflect point X. Point X is 3 units below the -ais. ount 3 units above the -ais and plot point X. Reflect point Y. Point Y is 1 unit below the -ais. ount 1 unit above the -ais and plot point Y. Reflect point Z. - - X X' 3 3 Y Y' 1 1 Z' Z M Notes Houghton Mifflin Harcourt Publishing ompan STEP Point Z is units below the -ais. ount units above the -ais and plot point Z. onnect X, Y, and Z to form triangle X Y Z. Each verte of the image is the same distance from the -ais as the corresponding verte in the original figure. - X' X Y Y' Z' - Z Lesson 9. 89

14 Personal Math Trainer nline Practice and Help m.hrw.com YUR TURN. The figure shows pentagon E. Graph the image of the pentagon after a reflection across the -ais. - E - Guided Practice 1. Vocabular reflection is a transformation that flips a figure across a line called the.. The figure shows trapezoid. (Eplore ctivities 1 and and Eample 1) a. Graph the image of the trapezoid after a reflection across the -ais. Label the vertices of the image. b. How do trapezoid and trapezoid '''' compare? c. What If? Suppose ou reflected trapezoid across the -ais. How would the orientation of the image of the trapezoid compare with the orientation of the preimage? -? ESSENTIL QUESTIN HEK-IN 3. What are the properties of reflections? - Houghton Mifflin Harcourt Publishing ompan 90 Unit

15 Name lass ate Houghton Mifflin Harcourt Publishing ompan 9. Independent Practice 8.G.1a, 8.G.1b, 8.G.1c, 8.G.3 The graph shows four right triangles. Use the graph for Eercises Which two triangles are reflections of each other across the -ais?. For which two triangles is the line of reflection the -ais?. Which triangle is a translation of triangle? How would ou describe the translation? 7. Which triangles are congruent? How do ou know? - W - m.hrw.com X Personal Math Trainer nline Practice and Help 8. a. Graph quadrilateral WXYZ with vertices W(-, -), X(3, 1), Y(, -1), and Z(, -) on the coordinate grid. Z Y b. n the same coordinate grid, graph quadrilateral W X Y Z, the image of quadrilateral WXYZ after a reflection across the -ais. c. Which side _ of the image is congruent to side YZ? Name three other pairs of congruent sides. d. Which angle of the image is congruent to X? Name three other pairs of congruent angles. Lesson 9. 91

16 9. ritical Thinking Is it possible that the image of a point after a reflection could be the same point as the preimage? Eplain. FUS N HIGHER RER THINKING 10. a. Graph the image of the figure shown after a reflection across the -ais. b. n the same coordinate grid, graph the image of the figure ou drew in part a after a reflection across the -ais. c. Make a onjecture What other sequence of transformations would produce the same final image from the original preimage? heck our answer b performing the transformations. Then make a conjecture that generalizes our findings a. Graph triangle EF with vertices (, ), E(, ), and F(, 1) on the coordinate grid. E b. Net graph triangle E F, the image of triangle EF after a reflection across the -ais. c. n the same coordinate grid, graph triangle E F, the image of triangle E F after a translation of 7 units down and units to the right. d. nalze Relationships Find a different sequence of transformations that will transform triangle EF to triangle E F F Houghton Mifflin Harcourt Publishing ompan 9 Unit

17 ? LESSN 9.3 Properties of Rotations ESSENTIL QUESTIN 8.G.1c Verif eperimentall the properties of rotations, reflections, and translations: Parallel lines are taken to parallel lines. lso 8.G.1a, 8.G.1b, 8.G.3 How do ou describe the properties of rotation and their effect on the congruence and orientation of figures? EXPLRE TIVITY 1 8.G.1c Eploring Rotations rotation is a transformation that turns a figure around a given point called the center of rotation. The image has the same size and shape as the preimage. The triangle shown on the grid is the preimage. You will use the origin as the center of rotation. Trace triangle onto a piece of paper. ut out our traced triangle. Rotate our triangle 90 counterclockwise about the origin. The side of the triangle that lies along the -ais should now lie along the -ais. Sketch the image of the rotation. Label the images of points,, and as,, and. - E escribe the motion modeled b the rotation. Rotate about the origin. degrees heck that the motion ou described in is the same motion that maps point onto, point onto, and point onto. Reflect 1. ommunicate Mathematical Ideas How are the size and the orientation of the triangle affected b the rotation? -. Rotate triangle 90 clockwise about the origin. Sketch the result on the coordinate grid above. Label the image vertices,, and. Lesson

18 EXPLRE TIVITY Properties of Rotations Use trapezoid TRP to investigate the properties of rotations. Trace the trapezoid onto a piece of paper. Include the portion of the - and -aes bordering the third quadrant. ut out our tracing. Place our trapezoid and aes on top of those in the figure. Then use the aes to help rotate our trapezoid 180 counterclockwise about the origin. Sketch the image of the rotation of our trapezoid in this new location. Label the vertices of the image T, R,, and P. Use a ruler to measure the sides of trapezoid TRP in centimeters. TR = R = P = TP = 8.G.1c - R - T P ut along line after tracing. Use a ruler to measure the sides of trapezoid T R P in centimeters. T R = R = P = T P = E What do ou notice about the lengths of corresponding sides of the two figures? F Use a protractor to measure the angles of trapezoid TRP. G H I m T = m R = m = m P = Use a protractor to measure the angles of trapezoid T R P. m T = m R = m = m P = What do ou notice about the measures of corresponding angles of the two figures? Which sides of trapezoid TRP are parallel? Which sides of trapezoid T R P are parallel? What do ou notice? Houghton Mifflin Harcourt Publishing ompan 9 Unit

19 Reflect 3. Make a onjecture Use our results from E, H, and I to make a conjecture about rotations.. Place our tracing back in its original position. Then perform a 180 clockwise rotation about the origin. ompare the result with the result in. Graphing Rotations To rotate a figure in the coordinate plane, rotate each of its vertices. Then connect the vertices to form the image. EXMPLE 1 8.G.3 Math n the Spot m.hrw.com The figure shows triangle. Graph the image of triangle after a rotation of 90 clockwise. STEP 1 Rotate the figure clockwise from the -ais to the -ais. Point will still be at (0, 0). - - nimated Math m.hrw.com Houghton Mifflin Harcourt Publishing ompan STEP Point is units to the left of the -ais, so point is units above the -ais. Point is units to the right of the -ais, so point is units below the -ais. onnect,, and to form the image triangle. Reflect. Is the image congruent to the preimage? How do ou know? Math Talk Mathematical Practices How is the orientation of the triangle affected b the rotation? - - Lesson 9.3 9

20 Personal Math Trainer nline Practice and Help m.hrw.com YUR TURN Graph the image of quadrilateral after each rotation clockwise 8. Find the coordinates of Point after a 90 counterclockwise rotation followed b a 180 rotation. - - Guided Practice 1. Vocabular rotation is a transformation that turns a figure around a given called the center of rotation. Siobhan rotates a right triangle 90 counterclockwise about the origin.. How does the orientation of the image of the triangle compare with the orientation of the preimage? (Eplore ctivit 1) 3. Is the image of the triangle congruent to the preimage? (Eplore ctivit ) raw the image of the figure after the given rotation about the origin. (Eample 1). 90 counterclockwise. 180? E - F G - ESSENTIL QUESTIN HEK-IN. What are the properties of rotations? - - Houghton Mifflin Harcourt Publishing ompan 9 Unit

21 Name lass ate 9.3 Independent Practice 8.G.1a, 8.G.1b, 8.G.1c, 8.G.3 m.hrw.com Personal Math Trainer nline Practice and Help 7. The figure shows triangle and a rotation of the triangle about the origin. a. How would ou describe the rotation? ' ' - ' b. What are the coordinates of the image?,, 8. The graph shows a figure and its image after a transformation. a. How would ou describe this as a rotation? b. an ou describe this as a transformation other than a rotation? Eplain. 9. What tpe of rotation will preserve the orientation of the H-shaped figure in the grid? 3 Houghton Mifflin Harcourt Publishing ompan 10. point with coordinates (-, -3) is rotated 90 clockwise about the origin. What are the coordinates of its image? omplete the table with rotations of 180 or 90. Include the direction of rotation for rotations of 90. Shape in quadrant Image in quadrant Rotation I III IV IV I III Lesson

22 raw the image of the figure after the given rotation about the origin counterclockwise Is there a rotation for which the orientation of the image is alwas the same as that of the preimage? If so, what? FUS N HIGHER RER THINKING Work rea 17. Problem Solving Lucas is plaing a game where he has to rotate a figure for it to fit in an open space. Ever time he clicks a button, the figure rotates 90 degrees clockwise. How man times does he need to click the button so that each figure returns to its original orientation? Figure Figure Figure 18. Make a onjecture Triangle is reflected across the -ais to form the image. Triangle is then reflected across the -ais to form the image. What tpe of rotation can be used to describe the relationship between triangle and triangle? 19. ommunicate Mathematical Ideas Point is on the -ais. escribe all possible locations of image for rotations of 90, 180, and 70. Include the origin as a possible location for. Houghton Mifflin Harcourt Publishing ompan 98 Unit

23 ? LESSN 9. lgebraic ESSENTIL QUESTIN lgebraic Representations of Translations The rules shown in the table describe how coordinates change when a figure is translated up, down, right, and left on the coordinate plane. Translations Representations of Transformations Right a units dd a to the -coordinate: (, ) ( + a, ) How can ou describe the effect of a translation, rotation, or reflection on coordinates using an algebraic representation? Left a units Subtract a from the -coordinate: (, ) ( - a, ) Up b units dd b to the -coordinate: (, ) (, + b) 8.G.3 escribe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Math n the Spot m.hrw.com own b units Subtract b from the -coordinate: (, ) (, - b) EXMPLE 1 8.G.3 Triangle XYZ has vertices X(0, 0), Y(, 3), and Z(, -1). Find the vertices of triangle X Y Z after a translation of 3 units to the right and 1 unit down. Then graph the triangle and its image. dd 3 to the -coordinate of each verte and subtract 1 from the STEP 1 ppl the rule to find the vertices of the image. -coordinate of each verte. Vertices of XYZ Rule: ( + 3, - 1) Vertices of X Y Z Houghton Mifflin Harcourt Publishing ompan STEP X(0, 0) (0 + 3, 0-1) X (3, -1) Y(, 3) ( + 3, 3-1) Y (, ) Z(, -1) ( + 3, -1-1) Z (7, -) Graph triangle XYZ and its image. 3 Y Y X Z 7 X -3 Z Math Talk Mathematical Practices When ou translate a figure to the left or right, which coordinate do ou change? Lesson 9. 99

24 YUR TURN Personal Math Trainer nline Practice and Help m.hrw.com 1. rectangle has vertices at (0, -), (0, 3), (3, -), and (3, 3). What are the coordinates of the vertices of the image after the translation (, ) ( -, - 3)? escribe the translation. Math n the Spot m.hrw.com lgebraic Representations of Reflections The signs of the coordinates of a figure change when the figure is reflected across the -ais and -ais. The table shows the rules for changing the signs of the coordinates after a reflection. Reflections cross the -ais Multipl each -coordinate b -1: (, ) (, -) cross the -ais Multipl each -coordinate b -1: (, ) (-, ) EXMPLE 8.G.3 M Notes Rectangle RSTU has vertices R(-, -1), S(-1, -1), T(-1, -3), and U(-, -3). Find the vertices of rectangle R S T U after a reflection across the -ais. Then graph the rectangle and its image. Multipl the -coordinate of STEP 1 ppl the rule to find the vertices of the image. each verte b -1. STEP Vertices of RSTU Rule: (-1, ) Vertices of R S T U R(-, -1) (-1 (-), - 1) R (, -1) S(-1, -1) (-1 (-1), - 1) S (1, -1) T(-1, -3) (-1 (-1), - 3) T (1, -3) U(-, -3) (-1 (-), - 3) U (, -3) Graph rectangle RSTU and its image. 3 Houghton Mifflin Harcourt Publishing ompan - R S S R U T T U Unit

25 YUR TURN. Triangle has vertices (-, ), (0, ), and (3, -1). Find the vertices of triangle after a reflection across the -ais. Personal Math Trainer nline Practice and Help m.hrw.com lgebraic Representations of Rotations When points are rotated about the origin, the coordinates of the image can be found using the rules shown in the table. Rotations 90 clockwise 90 counterclockwise Multipl each -coordinate b -1; then switch the - and -coordinates: (, ) (, -) Multipl each -coordinate b -1; then switch the - and -coordinates: (, ) (-, ) 180 Multipl both coordinates b -1: (, ) (-, -) Math n the Spot m.hrw.com EXMPLE 3 8.G.3 Quadrilateral has vertices at (-, ), (-3, ), (, 3), and (0, 0). Find the vertices of quadrilateral after a 90 clockwise rotation. Then graph the quadrilateral and its image. Multipl the -coordinate of each verte b -1, and then STEP 1 ppl the rule to find the vertices of the image. switch the - and -coordinates. Vertices of Rule: (, -) Vertices of (-, ) (, -1 (-)) (, ) (-3, ) (, -1 (-3)) (, 3) (, 3) (3, -1 ) (3, -) (0, 0) (0, -1 0) (0, 0) Houghton Mifflin Harcourt Publishing ompan STEP Graph the quadrilateral and its image Lesson

26 Reflect 3. ommunicate Mathematical Ideas How would ou find the vertices of an image if a figure were rotated 70 clockwise? Eplain. YUR TURN Personal Math Trainer nline Practice and Help m.hrw.com. triangle has vertices at J(-, -), K(1, ), and L(, ). What are the coordinates of the vertices of the image after the triangle is rotated 90 counterclockwise? Guided Practice 1. Triangle XYZ has vertices X(-3, -), Y(-1, 0), and Z(1, -). Find the vertices of triangle X Y Z after a translation of units to the right. Then graph the triangle and its image. (Eample 1) -3 Y 3 7. escribe what happens to the - and -coordinates after a point is reflected across the -ais. (Eample ) X -7 Z? 3. Use the rule (, ) (, -) to graph the image of the triangle at right. Then describe the transformation. (Eample 3) ESSENTIL QUESTIN HEK-IN. How do the - and -coordinates change when a figure is translated right a units and down b units? - -3 Houghton Mifflin Harcourt Publishing ompan 30 Unit

27 Name lass ate 9. Independent Practice 8.G.3 m.hrw.com Personal Math Trainer nline Practice and Help Write an algebraic rule to describe each transformation. Then describe the transformation... M N P - - M N P Triangle XYZ has vertices X(, -.3), Y(7., ), and Z(8, ). When translated, X has coordinates (.8, -1.3). Write a rule to describe this transformation. Then find the coordinates of Y and Z. 8. Point L has coordinates (3, -). The coordinates of point L after a reflection are (-3, -). Without graphing, tell which ais point L was reflected across. Eplain our answer. Houghton Mifflin Harcourt Publishing ompan 9. Use the rule (, ) ( -, - ) to graph the image of the rectangle. Then describe the transformation Parallelogram has vertices (-, - 1_ ), (-, - 1_ ), (-3, -), and (-1, -). Find the vertices of parallelogram after a translation of 1_ units down. - Lesson

28 11. leandra drew the logo shown on half-inch graph paper. Write a rule that describes the translation leandra used to create the shadow on the letter. 1. Kite KLMN has vertices at K(1, 3), L(, ), M(3, 3), and N(, 0). fter the kite is rotated, K has coordinates (-3, 1). escribe the rotation, and include a rule in our description. Then find the coordinates of L, M, and N. FUS N HIGHER RER THINKING 13. Make a onjecture Graph the triangle with vertices (-3, ), (3, ), and (-, -). Use the transformation (, ) to graph its image. a. Which verte of the image has the same coordinates as a verte of the original figure? Eplain wh this is true. - b. What is the equation of a line through the origin and this point? - c. escribe the transformation of the triangle. 1. ritical Thinking Mitchell sas the point (0, 0) does not change when reflected across the - or -ais or when rotated about the origin. o ou agree with Mitchell? Eplain wh or wh not. Work rea 1. nalze Relationships Triangle with vertices (-, -), (-3, 1), and (1, 1) is translated b (, ) ( - 1, + 3). Then the image, triangle, is translated b (, ) ( +, - 1), resulting in. Houghton Mifflin Harcourt Publishing ompan a. Find the coordinates for the vertices of triangle. b. Write a rule for one translation that maps triangle to triangle. 30 Unit

29 ? L E S S N 9. ongruent Figures ESSENTIL QUESTIN How can transformations be used to verif that two figures have the same shape and size? 8.G. Understand that a twodimensional figure is congruent to another if the second can be obtained from the first b a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that ehibits the congruence between them. EXPLRE TIVITY 8.G. ombining Transformations ppl the indicated series of transformations to the triangle. Each transformation is applied to the image of the previous transformation, not the original figure. Label each image with the letter of the transformation applied. E F Reflection across the -ais (, ) ( - 3, ) Reflection across the -ais (, ) (, + ) Rotation 90 clockwise around the origin ompare the size and shape of the final image to that of the original figure. - - Houghton Mifflin Harcourt Publishing ompan Reflect 1. Which transformation(s) change the orientation of figures? Which do not?. Make a onjecture fter a series of transformations, two figures have the same size and shape, but different orientations. What does this indicate about the transformations? Lesson 9. 30

30 Math n the Spot m.hrw.com ongruent Figures Recall that segments and their images have the same length and angles and their images have the same measure under a translation, reflection, or rotation. Two figures are said to be congruent if one can be obtained from the other b a sequence of translations, reflections, and rotations. ongruent figures have the same size and shape. When ou are told that two figures are congruent, there must be a sequence of translations, reflections, and/or rotations that transforms one into the other. EXMPLE 1 8.G. Identif a sequence of transformations that will transform figure into figure. - - Math Talk Mathematical Practices How do ou know that the sequence of transformations in Parts and must include a rotation? To transform figure into figure, ou need to reflect it over the -ais and translate one unit to the left. sequence of transformations that will accomplish this is (, ) (-, ) and (, ) ( - 1, ). Identif a sequence of transformations that will transform figure into figure n sequence of transformations that changes figure into figure will need to include a rotation. 90 counterclockwise rotation around the origin would properl orient figure, but not locate it in the same position as figure. The rotated figure would be units below and 1 unit to the left of where figure is. You would need to translate the rotated figure up units and right 1 unit. sequence of transformations is a 90 counterclockwise rotation about the origin, (, ) (-, ), followed b (, ) ( + 1, + ). Houghton Mifflin Harcourt Publishing ompan 30 Unit

31 Identif a sequence of transformations that will transform figure into figure E. M Notes E - sequence of transformations that changes figure to figure E will need to include a rotation. 90º clockwise rotation around the origin would result in the figure being oriented as figure E. However, the rotated figure would be units above where figure E is. You would need to translate the rotated figure down units. sequence of transformations is a 90º clockwise rotation about the origin, (, ) (, -), followed b (, ) (, - ). YUR TURN Houghton Mifflin Harcourt Publishing ompan 3. Identif a sequence of transformations that will transform figure into figure Personal Math Trainer nline Practice and Help m.hrw.com Lesson

32 Guided Practice 1. ppl the indicated series of transformations to the rectangle. Each transformation is applied to the image of the previous transformation, not the original figure. Label each image with the letter of the transformation applied. (Eplore ctivit). Reflection across the -ais. Rotation 90 clockwise around the origin. (, ) ( -, ). Rotation 90 counterclockwise around the origin E. (, ) ( - 7, - ) Identif a sequence of transformations that will transform figure into figure. (Eample 1) What transformation is used to transform figure into figure? 3. What transformation is used to transform figure into figure? - -. What sequence of transformations is used to transform figure into figure? Epress the transformations algebraicall.. Vocabular What does it mean for two figures to be congruent?? ESSENTIL QUESTIN HEK-IN. fter a sequence of translations, reflections, and rotations, what is true about the first figure and the final figure? Houghton Mifflin Harcourt Publishing ompan 308 Unit

33 Name lass ate 9. Independent Practice 8.G. m.hrw.com Personal Math Trainer nline Practice and Help For each given figure, graph figures and using the given sequence of transformations. State whether figures and have the same or different orientation Figure : a translation of 1 unit to the right and 3 units up Figure : a 90 clockwise rotation around the origin Figure : a reflection across the -ais Figure : a 180 rotation around the origin Houghton Mifflin Harcourt Publishing ompan Figure : a reflection across the -ais Figure : a translation units up Figure : a translation units down Figure : a rotation of 180 around the origin Lesson

34 11. Represent Real-World Problems cit planner wanted to place the new town librar at site. The maor thought that it would be better at site. What transformations were applied to the building at site to relocate the building to site? id the maor change the size or orientation of the librar? Site Site 1. Persevere in Problem Solving Find a sequence of three transformations that can be used to obtain figure from figure. Graph the figures and that are created b the transformations FUS N HIGHER RER THINKING Work rea 13. ountereamples The ommutative Properties for ddition and Multiplication state that the order of two numbers being added or multiplied does not change the sum or product. re translations and rotations commutative? If not, give a countereample. 1. Multiple Representations For each representation, describe a possible sequence of transformations. a. (, ) (- -, + 1) Houghton Mifflin Harcourt Publishing ompan b. (, ) (, - - 3) 310 Unit

35 MULE QUIZ Read Properties of Translations, Reflections, and Rotations Personal Math Trainer nline Practice and Help m.hrw.com 1. Graph the image of triangle after a translation of units to the right and units down. Label the vertices of the image,, and.. n the same coordinate grid, graph the image of triangle after a reflection across the -ais. Label the vertices of the image,, and Graph the image of HIJK after it is rotated 180 about the origin. Label the vertices of the image H I J K lgebraic Representations of Transformations K H. triangle has vertices at (, 3), (, ), and ( 3, ). What are the coordinates of the vertices of the image after the translation (, ) ( +, 3)? - J I 9. ongruent Figures. Vocabular Translations, reflections, and rotations produce a figure - that is to the original figure. Houghton Mifflin Harcourt Publishing ompan. Use the coordinate grid for Eercise 3. Reflect H I J K over the -ais, then rotate it 180 about the origin. Label the new figure H I J K. ESSENTIL QUESTIN 7. What properties allow transformations to be used as problem solving tools? Module 9 311

36 MULE 9 MIXE REVIEW ssessment Readiness m.hrw.com Personal Math Trainer nline Practice and Help 1. onsider each rational number. Is the number greater than 1_ 3 but less than _? Select Yes or No for.. 0. Yes No. 9_ 7 Yes No. 0.9 Yes No. Triangle EF is rotated 90 clockwise about the origin. hoose True or False for each statement.. E F = 3 True False. m F = 90 True False _. F is vertical. True False 3. graphic artist is working on a logo for a car compan. The artist draws parallelogram LMNP with vertices L(1, 0), M(, ), N(, 3), and P(, 1). The artist then reflects the parallelogram across the -ais. Find the coordinates of the vertices of parallelogram L M N P, and describe the algebraic rule ou used to find the coordinates F E. The map shows two of a farmer s fields. Is field congruent to field? Use a sequence of transformations to eplain our reasoning Houghton Mifflin Harcourt Publishing ompan 31 Unit