Topic 8 Transformational Geometry
TOPIC OVERVIEW VOCABULARY
8-1 Translations English/Spanish Vocabulary Audio Online: 8-2 Reflections English Spanish 8-3 Rotations compression, p. 364 compreción 8-4 Symmetry congruence transformation, p. 350 transformación de congruencia 8-5 Compositions of Rigid dilation, p. 356 dilatación Transformations image, p. 318 imagen preimage, p. 318 preimagen 8-6 Congruence Transformations reflection, p. 326 reflexión 8-7 Dilations rigid transformation, p. 318 transformación rígido 8-8 Other Non-Rigid Transformations stretch, p. 364 estiramiento rotation, p. 332 rotación translation, p. 319 translación
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317 8-1 Translations
TEKS FOCUS VOCABULARY TEKS (3)(A) Describe and perform • Composition of transformations – A composition of transformations transformations of figures in a plane using is a combination of two or more transformations. In a composition, coordinate notation. you perform each transformation on the image of the preceding transformation. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications • Image – the resulting figure in a transformation using multiple representations, including • Preimage – the original figure in a transformation symbols, diagrams, graphs, and language as • Rigid transformation – a transformation that preserves distance and appropriate. angle measures Additional TEKS (1)(F), (3)(C), (6)(C) • Transformation – a function, or mapping, that results in a change in the position, shape, or size of a figure • Translation – a transformation that maps all points of a figure the same distance in the same direction.
• Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated • Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING You can change the position of a geometric figure so that the angle measures and the distance between any two points of a figure stay the same.
Key Concept Transformations
A transformation is a function that maps every point of a figure, called the preimage, onto its image. A transformation may be described with arrow notation (S). Prime notation (′) is sometimes used to identify image points. In the diagram below, K ′ is the image of K. J JЈ ᭝JKQ S ᭝JЈKЈQЈ K KЈ ᭝JKQ maps onto ᭝JЈKЈQЈ.
Q QЈ
Notice that you list corresponding points of the preimage and image in the same order, as you do for corresponding points of congruent figures.
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318 Lesson 8-1 Translations Key Concept Translation
A translation is a transformation that maps all points of a BЈ B figure the same distance in the same direction. You write the translation that maps △ABC onto △A′B′C′ using A AЈ the function notation T (△ABC) = △A′B′C′. A translation is a CЈ rigid transformation with the following properties. C If T (△ABC) = △A′B′C′, then • AA′ = BB′ = CC′ • AB = A′B′, BC = B′C′, AC = A′C′ • m∠A = m∠A′, m∠B = m∠B′, m∠C = m∠C′ hsm11gmse_0901_t07523.ai
Key Concept Translation in the Coordinate Plane
A translation can be performed as a composition of a AB y horizontal and a vertical translation. In the diagram at the B moves 4 units 2 right, each point of ABCD is translated 4 units right and AЈ BЈ right and 2 units down. So each (x, y) pair in ABCD is mapped to DC x 2 units down. Ϫ O (x + 4, y - 2). You can use the function notation 2 Ϫ2 DЈ CЈ T64, -27 (ABCD) = A′B′C′D′ to describe this translation, where 4 represents the horizontal translation of each point of the figure and -2 represents the vertical translation. T64, -27 (x, y) = (x + 4, y - 2) (x, y) S (x + 4, y - 2)
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Problem 1 Identifying a Rigid Transformation What must be true about a rigid Does the transformation below appear to be a rigid transformation? Explain. transformation? In a rigid transformation, the image and the preimage must preserve distance and angle measures.
Preimage Image No, a rigid transformation preserves both distance and angle measure. In this transformation, the distances between the vertices of the image are not the same as the corresponding distances in the preimage. hsm11gmse_0901_t07517.ai
PearsonTEXAS.com 319 Problem 2 TEKS Process Standard (1)(F) Naming Images and Corresponding Parts In the diagram, EFGH E F G H . FЈ How do you identify u corresponding A What are the images of jF and jH? GЈ points? HЈ F is the image of F. H is the image of H. F Corresponding points ∠ ′ ∠ ∠ ′ ∠ EЈ have the same position B What are the pairs of corresponding sides? G in the names of the H preimage and image. You EF and E′F ′ FG and F ′G′ can use the statement E EFGH S E′F′G′H′. EH and E′H′ GH and G′H′ EFGH S EЈFЈGЈHЈ
Problem 3 hsm11gmse_0901_t07521.ai Finding the Image of a Translation
What are the vertices of T*2, 5+(△PQR)? Graph the image of △PQR. y 4 R Q
P x What does the rule O tell you about the Ϫ4 Ϫ2 2 4 direction each point Identify the coordinates of each vertex. Use the coordinate rule moves? T (x, y) (x 2, y 5) to find the coordinates of each vertex of the image. -2 means that each 6-2, -57 = - - point moves 2 units left. T 2, 5 (P) = (2 - 2, 1 - 5), or P′(0, -4). -5 means that each 6- - 7 hsm11gmse_0901_t07525.ai point moves 5 units T6-2, -57(Q) = (3 - 2, 3 - 5), or Q′(1, -2). down. T6-2, -57(R) = (-1 - 2, 3 - 5), or R′(-3, -2). To graph the image of △PQR, first graph P′, Q′, and R′. Then draw P′Q′, Q′R′, and R′P′.
y 4 R Q
P x Ϫ4 Ϫ2 O 2 4 RЈ QЈ
PЈ
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320 Lesson 8-1 Translations Problem 4 TEKS Process Standard (1)(D) Writing a Rule to Describe a Translation What is a coordinate rule that describes the translation that maps PQRS onto P Q R S ? P y 4 PЈ S 2 Q SЈ x Ϫ Ϫ O 6 2 4 QЈ R Ϫ2 RЈ
The coordinates of An algebraic relationship that Use one pair of corresponding vertices to the vertices of both maps each point of PQRS find the change in the horizontal direction figures hsm11gmse_0901_t07528.aionto P′Q′R′S′ x and the change in the vertical direction y. Then use the other vertices to verify.
Use P(Ϫ3, 4) and its image PЈ(5, 2). Horizontal change: 5 Ϫ (Ϫ3) ϭ 8 x S x ϩ 8 How do you know which pair of P(Ϫ3, 4) y corresponding Vertical change: 2 Ϫ 4 ϭ Ϫ2 vertices to use? S 2 PЈ(5, 2) y S y Ϫ 2 A translation moves all Q points the same distance SЈ x Ϫ Ϫ O and the same direction. 6 2 4 QЈ You can use any pair of R Ϫ2 corresponding vertices. RЈ
The translation maps each (x, y) to (x + 8, y - 2). The coordinate rule that describes the translation is T68, -27(x, y) = (x + 8, y - 2). hsm11gmse_0901_t08340.ai
PearsonTEXAS.com 321 Problem 5 Composing Translations Chess The diagram at the right shows two moves of the black bishop in a chess game. Where is the bishop in relation to its original position? 1
2
How can you define the bishop’s original Use (0, 0) to represent the bishop’s original position. Write coordinate rules to position? represent each move. You can think of the chessboard as a T64, -47(x, y) = (x + 4, y - 4) The bishop moves 4 squares right and 4 squares down. coordinate plane with T (x, y) (x 2, y 2) The bishop moves 2 squares right and 2 squares up. the bishop’s original 62, 27 = + + position at the origin. The bishop’s current position is the composition of the two translations.
First, T64, -47(0, 0) = (0 + 4, 0 - 4), or (4, -4).
Then T62, 27(4, -4) = (4 + 2, -4 + 2), or (6, -2). The bishop is 6 squares right and 2 squares down from its original position.
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Tell whether the transformation appears to be a rigid transformation. Explain.
1. Image 2. 3. For additional support when Image completing your homework, Preimage Preimage Image go to PearsonTEXAS.com. Preimage
4. You are a graphic designer for a company that manufactures wrapping paper. Make a design for wrapping paper that involves translations. hsm11gmse_0901_t06719.ai 5. hsm11gmse_0901_t06718.aiAnalyze Mathematical Relationships (1)(F) Your friend andhsm11gmse_0901_t06720.ai her parents are visiting colleges. They leave their home in Enid, Oklahoma, and drive to Tulsa, which is 107 mi east and 18 mi south of Enid. From Tulsa, they go to Norman, 83 mi west and 63 mi south of Tulsa. Where is Norman in relation to Enid?
322 Lesson 8-1 Translations In each diagram, the blue figure is an image of the black figure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides.
6. QR 7. PЈ RЈ R 8. G W R M
TЈ P T
RЈ P S SЈ B N X S TP P Ј QЈ 9. In the diagram at the right, the orange figure is a translation image of the red figure. Write a coordinate rule that describes the translation.hsm11gmse_0901_t06723.ai 10. Displayhsm11gmse_0901_t06721.ai Mathematical Ideas (1)(G)hsm11gmse_0901_t06722.ai △MUG has coordinates M(2, -4), U(6, 6), and G(7, 2). A translation maps point M to M′( -3, 6). What are the coordinates of U′ and G′ for this translation? 11. Justify Mathematical Arguments (1)(G) PLAT has vertices P( -2, 0), L( -1, 1), A(0, 1), and T( -1, 0). The translation T62, -37(PLAT) = P′L′A′T′. Show that PP′, LL′, AA′, and TT ′ are all parallel. hsm11gmse_0901_t06733.ai 12. Analyze Mathematical Relationships (1)(F) If T65, 77(△MNO) = △M′N′O′, what coordinate rule maps △M′N′O′ onto △MNO? 13. Apply Mathematics (1)(A) The diagram at the right shows the site plan for a backyard storage shed. Local law, however, requires the shed to sit at N least 15 ft from property lines. Describe how to move the shed to comply with the law. 10 ft
STEM 14. You write a computer animation program to help young children learn
Property Line 5 ft the alphabet. The program draws a letter, erases the letter, and makes it Property Line reappear in a new location two times. The program uses the following composition of translations to move the letter.
T65, 77(x, y) followed by T6-9, -27(x, y) Suppose the program makes the letter W by connecting the points (1, 2), (2, 0), (3, 2), (4, 0) and (5, 2). What points does the program connect to make the last W? 15. Connect Mathematical Ideas (1)(F) △ABC has vertices A( -2, 5), B( -4, -1), and C (2, -3). If T64, 27(△ABC) = △A′B′C′, show that the images of the midpoints of the sides of △ABC are the midpoints of the sides of △A′B′C ′. 16. Explain Mathematical Ideas (1)(G) Explain how to use translations to draw a parallelogram. J y 17. Use the graph at the right. Write three different rules for which the 4 image of △JKL has a vertex at the origin. 2 K L x Ϫ4 Ϫ2 O 2
PearsonTEXAS.com 323 hsm11gmse_0901_t06734.ai Find a translation that has the same effect as each composition of translations.
18. T62, 57(x, y) followed by T6-4, 97(x, y)
19. T612, 0.57(x, y) followed by T61, -37(x, y) Copy each graph. Graph the image of each figure under the given translation.
20. T63, 27(x, y) 21. T6-2, 57(x, y)
y y 3 2 x x Ϫ6 Ϫ2 O 2 Ϫ8 Ϫ4 Ϫ2 O Ϫ2 Ϫ3
The blue figure is a translation image of the black figure. Write coordinate rules to describe each translation. hsm11gmse_0901_t06724.ai 22. y 23. hsm11gmse_0901_t06726.aiy 4 6
2 4 x Ϫ2 O x Ϫ2 O 64
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24. △ABC has vertices A( -5, 2), B(0, -4), and C(3, 3). What are the vertices of the image of △ABC after the translation T67, -57(△ABC)? A. A′(2, -3), B′(7, -9), C′(10, -2) C. A′( -12, 7), B′( -7, 1), C′( -4, 8) B. A′( -12, -3), B′( -7, -9), C′( -4, -2) D. A′(2, -3), B′(10, -2), C′(7, -9) 25. In △PQR, PQ = 4.5, QR = 4.4, and RP = 4.6. Which statement is true? F. m∠P + m∠Q 6 m∠R H. ∠R is the largest angle. G. ∠Q is the largest angle. J. m∠R 6 m∠P 26. ▱ABCD has vertices A(0, -3), B( -4, -2), and D( -1, 1). Point C is in Quadrant II. a. What are the coordinates of C? b. Is ▱ABCD a rhombus? Explain.
324 Lesson 8-1 Translations Activity Lab Paper Folding and Reflections
Use With Lesson 8-2 teks (3)(A), (1)(E)
In Activity 1, you will see how a figure and its reflection image are related. In Activity 2, you will use these relationships to construct a reflection image. 1
Step 1 Use a piece of tracing paper and a straightedge. Using less than half the page, draw a large, scalene triangle. Label its vertices A, B, and C. A Step 2 Fold the paper so that your triangle is covered. Trace △ABC using a straightedge. Step 3 Unfold the paper. Label the traced points corresponding to A, B, and C as C A′, B′, and C′, respectively. △A′B′C′ is a reflection image of △ABC. The fold B is the reflection line.
AЈ A AЈ A 0 1 2 3 4 5 76 hsm11gmse_0902a_t10066.ai CЈ C CЈ C BЈ B BЈ B
1. Use a ruler to draw AA′. Measure the perpendicular distances from A to the fold and from A′ to the fold. What do you notice? 2. Measure thehsm11gmse_0902a_t08779.ai angles formed by the fold and AA′. What are the angle measures? 3. Repeat Exercises 1 and 2 for B and B′and for C andhsm11gmse_0902a_t08780.ai C′. Then, make a conjecture: How is the reflection line related to the segment joining a point and its image? 2
Step 1 On regular paper, draw a simple shape or design made of segments. Use less than half the page. Draw a reflection line near your figure. Step 2 Use a compass and straightedge to construct a perpendicular to the DЈ reflection line through one point of your drawing. D
4. Explain how you can use a compass and the perpendicular you drew to find the reflection G E reflection image of the point you chose. line S F 5. Connect the reflection images for several points of your shape and complete the image. Check the accuracy of the reflection image by folding the paper along the reflection line and holding it up to a light source.
hsm11gmse_0902a_t08781.ai PearsonTEXAS.com 325 8-2 Reflections
TEKS FOCUS VOCABULARY TEKS (3)(A) Describe and perform • Line of reflection – See reflection. transformations of figures in a • Orientation – the order in which the vertices of the figure appear in either a plane using coordinate notation. clockwise or counterclockwise order TEKS (1)(E) Create and use • Reflection – A reflection across a line m, called the line of reflection, is a representations to organize, transformation such that if a point A is on line m, then the image of A is itself, record, and communicate and if a point B is not on line m, then m is the perpendicular bisector of BB′. mathematical ideas.
Additional TEKS (1)(D), (1)(G), • Representation – a way to display or describe information. You can use a (3)(C) representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING When you reflect a figure across a line, each point of the figure maps to another point the same distance from the line but on the other side. The orientation of the figure reverses.
Key Concept Reflection Across a Line
A reflection across a line m, called the line of reflection, B The preimage B and is a transformation with the following properties: its image B’ are • If a point A is on line m, then the image of A is itself equidistant from (that is, A′ = A). C the line of reflection. • If a point B is not on line m, then m is the A perpendicular bisector of BB′. AЈ You write the reflection across m that takes △ABC to BЈ △A′B′C′ as Rm(△ABC) = △A′B′C′. m CЈ A reflection is a rigid transformation with the following properties: • Reflections preserve distance. If R (A) = A′, and R (B) = B′, then AB = A′B′. m m hsm11gmse_0902_t08439.ai • Reflections preserve angle measure. If Rm(∠ABC) = ∠A′B′C′, then m∠ABC = m∠A′B′C′. • Reflections map each point of the preimage to one and only one corresponding point of its image. Rm(A) = A′ if and only if Rm(A′) = A.
326 Lesson 8-2 Reflections Key Concept Reflection in the Coordinate Plane
Reflection across the x-axis Reflection across the y-axis y y Q(4, 3) 2 Q ( 4, 3) 2 Q(4, 3) x x -6 -4 -2 O 2 4 6 -6 -4 -2 O 2 4 6 -2 -2 Q (4, 3)
Multiply the y-coordinate by 21. Multiply the x-coordinate by 21.
Rx@axis(x, y) = (x, -y) Ry@axis(x, y) = (-x, y) (x, y) S (x, -y) (x, y) S (-x, y)
Problem 1
Reflecting a Point Across a Line
Multiple Choice Point P has coordinates (3, 4). What are the coordinates of Ry = 1(P)? (3, -4) (0, 4) (3, -2) (-3, -2) Graph point P and the line of reflection y = 1. P and its reflection image across the line must be equidistant from the line of reflection.
How does a graph y 4 help you visualize the P Move along the line through P that problem? 2 is perpendicular to the line of reflection. 1 ؍ A graph shows that y y = 1 is a horizontal line, x so the line through Ϫ2 O 2 4 P that is perpendicular to Stop when the distances of P and PЈ Ϫ2 the line of reflection is a PЈ to the line of reflection are the same. vertical line.
P is 3 units above the line y = 1, so P′ must be 3 units below the line y = 1. The line y = 1 is the perpendicular bisector of PP′ if P′ is (3, -2). The correct answer is C. hsm11gmse_0902_t08440.ai
PearsonTEXAS.com 327 Problem 2 TEKS Process Standard (1)(D) Graphing a Reflection Image
Graph points A(23, 4), B(0, 1), and C(4, 2). Graph and label Ry@axis(△ABC). Point B is located on the line of reflection. Step 1 Graph △ABC. Show the y-axis as the y 5 How will point B9 dashed line of reflection. A relate to the line of reflection? C Point B9 will also be on the line of reflection. B x Ϫ4 Ϫ2 O 2 4
Step 2 Find A′, B′, and C′ using the coordinate y 5 rule (x, y) S (-x, y). A AЈ
A(- 3, 4) S A′(3, 4) hsm11gmse_0902_t08441.aiCЈ C B(0, 1) S B′(0, 1) BЈ B x C(4, 2) S C′(-4, 2) Ϫ4 Ϫ2 O 2 4 Locate A′(3, 4), B′(0, 1), and C′(-4, 2) on the coordinate plane. Draw △A′B′C′.
Problem 3 TEKS Process Standard (1)(E) hsm11gmse_0902_t08443.ai Writing a Reflection Rule Each triangle in the diagram is a reflection of another triangle across one of the given lines. 1 How can you describe Triangle 2 by using a m reflection rule? If Triangle 2 is the 2 image of a reflection, Triangle 2 is the image of a reflection, so find the 3 what do you know preimage and the line of reflection to write a rule. about the preimage? The preimage has The preimage cannot be Triangle 3 because opposite orientation, and Triangle 2 and Triangle 3 have the same k 4 lies on the opposite side orientation and reflections reverse orientation. of the line of reflection. Check Triangles 1 and 4 by drawing line segments that connect the corresponding vertices of Triangle 2. Because neither line k nor line m is the perpendicular bisector of the segment drawn from Triangle 1 to hsm12_geo_se_t0001 1 Triangle 2, Triangle 1 is not the preimage. m Line k is the perpendicular bisector of the segments 2 3 joining corresponding vertices of Triangle 2 and Triangle 4. So Triangle 2 = Rk(Triangle 4).
k 4
328 Lesson 8-2 Reflections hsm12_geo_se_t0001A Problem 4 Using Properties of Reflections What do you have to In the diagram, R (G) G, R (H) J, and R (D) D. Use the properties of know about △GHJ t t t G to show that it is an reflections to describe how you know that △GHJ is an isosceles triangle. isosceles triangle? Isosceles triangles have Since Rt(G) = G, Rt(H) = J, and reflections preserve distance, Rt(GH ) = GJ. at least two congruent So GH = GJ and, by definition, △GHJ is an isosceles triangle. sides. H J D
t
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H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O hsm12_geo_se_t0002 Create Representations to Communicate Mathematical Ideas (1)(E) Given points J(1, 4), A(3, 5), and G(2, 1), graph △JAG and its reflection image as indicated.
For additional support when 1. Rx-axis 2. Ry-axis 3. Ry = 2 completing your homework, go to PearsonTEXAS.com. 4. Ry = 5 5. Rx = -1 6. Rx = 2
7. Each figure in the diagram at the right is a reflection Figure 3 of another figure across one of the reflection lines. j a. Write a reflection rule to describe Figure 3. Justify your answer. Figure 4 b. Write a reflection rule to describe Figure 2. Figure 1 Justify your answer. n Figure 2 c. Write a reflection rule to describe Figure 4. Justify your answer. 8. Apply Mathematics (1)(A) Give three examples from everyday life of objects or situations that show or use reflections. 9. In the diagram at the right, LMNP is a rectangle with LM = 2MN. LM
a. Copy the diagram. Then sketch R LM (LMNP). P N b. What figure results from the reflection? Use properties of reflectionsgeom12_se_ccs_t0003.ai to justify your solution. Copy each pair of figures. Then draw the line of reflection you can use to map one figure onto the other. hsm12_geo_se_t0004 10. 11.
hsm11gmse_0902_t06746.ai PearsonTEXAS.com 329
hsm11gmse_0902_t06745.ai 12. Explain Mathematical Ideas (1)(G) The following steps explain y x ؍ how to reflect point A across the line y = x. AЈ y Step 1 Draw line / through A(5, 1) perpendicular to the line 4 y = x. The slope of y = x is 1, so the slope of line / is 2 1 # (-1), or -1. A B x Step 2 From A, move two units left and two units up to O 5 y = x. Then move two more units left and two more Ϫ2 units up to find the location of A′ on line /. The ഞ coordinates of A′ are (1, 5). C a. Copy the diagram. Then draw the lines through B and C that are perpendicular to the line y = x. What is the slope of each line? b. R (B) = B′ and R (C) = C ′. What are the coordinates of B′ and C ′? y = x y = x hsm11gmse_0902_t14072 c. Graph △A′B′C′. d. Compare the coordinates of the vertices of △ABC and △A′B′C ′. Make a conjecture about the coordinates of the point P(a, b) reflected across the line y = x. 13. In the diagram R(ABCDE) = A′B′C′D′E′. What is the E y equation of the line of reflection? Write a coordinate rule D A that describes this reflection.
14. Use Representations to Communicate Mathematical C B Ideas (1)(E) The coordinates of the vertices of △FGH are AЈ F(2, -1), G(-2, -2), and H(-4, 3). Graph △FGH and 2 BЈ EЈ Ry = x - 3(△FGH). x 15. Use Multiple Representations to Communicate -4 -2O 2 DЈ CЈ Mathematical Ideas (1)(D) △ABC has vertices A(-3, 5), B(-2, -1), and C(0, 3). Graph Ry = -x(△ABC) and label it. 16. Explain Mathematical Ideas (1)(G) The work of artist and scientist Leonardo da Vinci (1452–1519) has an unusual characteristic. His handwriting is a mirror image of normal handwriting. a. Write the mirror image of the sentence “Leonardo da Vinci was left-handed.” Use a mirror to check how well you did. b. Explain why the fact about da Vinci in part (a) might have made mirror writing seem natural to him.
330 Lesson 8-2 Reflections 17. Display Mathematical Ideas (1)(G) Recall that when a ray of light hits a Mirrored wall mirror, it bounces off the mirror at the same angle at which it hits the mirror. You are installing a security camera. At what point on the mirrored wall should you aim the camera at C in order to view the door at D? Draw a C D diagram and explain your reasoning. 18. Explain Mathematical Ideas (1)(G) When you reflect a figure across a line, does every point on the preimage move the same distance? Explain. Find the coordinates of each image. hsm11gmse_0902_t06743.aiy 3 P S 19. Rx = 1(Q) 20. Ry = -1(P) 1 V x 21. Ry-axis(S) 22. Ry = 0.5(T) Ϫ2 O 2 4 23. Rx 3(U ) 24. Rx-axis(V) = - T Ϫ2 Q Explain Mathematical Ideas (1)(G) Can you form the given type of U quadrilateral by drawing a triangle and then reflecting one or more times? Explain. 25. parallelogram 26. isosceles trapezoid 27. kite 28. rhombus 29. rectangle 30. square
31. Show that Ry = x(A) = B for points A(a, b) and B(b, a). 32. Use the diagram at the right. Find the coordinates of each image point. y A (1, 3) a. R (A) = A′ 2 x ؍ y = x y x b. Ry = -x(A′) = A″ Ϫ2 2 c. Ry = x(A″) = A′″ Ϫ2 d. Ry x(A′″) = A″″ ؊x ؍ y - = e. How are A and A″″ related?
TEXAS Test Practice hsm11gmse_0902_t06753.ai
33. What is the reflection image of (a, b) across the line y = -6? A. (a - 6, b) C. (-12 - a, b) B. (a, b - 6) D. (a, -12 - b) 34. The diagonals of a quadrilateral are perpendicular and bisect each other. What is the most precise name for the quadrilateral? F. rectangle H. rhombus G. parallelogram J. kite 35. Write an indirect proof of the following statement: The hypotenuse of a right triangle is the longest side of the right triangle.
PearsonTEXAS.com 331 8-3 Rotations
TEKS FOCUS VOCABULARY TEKS (3)(C) Identify the sequence of • Angle of rotation – the positive number of degrees that a figure transformations that will carry a given rotates pre-image onto an image on and off the • Center of rotation – See rotation. coordinate plane. • Rotation – A rotation (turn) of x° about point Q, called the center of TEKS (1)(E) Create and use representations rotation, is a transformation such that for any point V, its image is the to organize, record, and communicate point V′, where QV′ = QV and m∠VQV′ = x. The image of Q is itself. mathematical ideas.
Additional TEKS (1)(D), (1)(F), (3)(A), • Representation – a way to display or describe information. You can use (6)(C) a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING Rotations preserve distance, angle measures, and orientation of figures.
Key Concept Rotation About a Point
A rotation of x° about a point Q, called the WЈ center of rotation, is a transformation with UЈ these two properties: VЈ The preimage V and • The image of Q is itself (that is, Q′ Q). = QЈ its image VЈ are • For any other point V, QV ′ = QV and x Њ Q equidistant from m∠VQV ′ = x. the center of rotation. The number of degrees a figure rotates is the V W angle of rotation. A rotation about a point is a rigid transformation. U You write the x° rotation of △UVW about point Q as r(x°, Q)(△UVW) = △U′V′W′. Unless stated otherwise, rotations in this course are counterclockwise. hsm11gmse_0903_t08068.ai
332 Lesson 8-3 Rotations Key Concept Rotation in the Coordinate Plane
r(90°, O)(x, y) = (-y, x) r(180°, O)(x, y) = (-x, -y) (x, y) S (-y, x) (x, y) S (-x, -y) y y 4 4 G(2, 3) G(2, 3) G (-3, 2) 2 2 x 1805 x -6 -4 -2 O 2 4 6 -6 -4 -2 2 4 6 -2 -2 G (-2, -3)
r(270°, O)(x, y) = (y, -x) r(360°, O)(x, y) = (x, y) (x, y) S (y, -x) (x, y) S (x, y) y y 4 4 G(2, 3) G (2, 3) 2 2 G(2, 3) x x -6 -4 -2 2 4 6 -6 -4 -2 42 6 2705 3605 -2 -2 G (3, -2)
Problem 1 TEKS Process Standard (1)(E) C Drawing a Rotation Image
What is the image of r(100, C)(△LOB)? O
L B How do you use the definition of rotation Step 1 Step 2 Step 3 Step 4 about a point to help Draw CO. Use a Use a compass to Locate B9 and L9 in a Draw △L′O′B′. you get started? You know that O and O9 protractor to draw a construct CO′ ≅ CO. similarhsm11gmse_0903_t08070.ai manner. must be equidistant from 100° angle with C and that m∠OCO′ vertex C and side CO. BЈ must be 100. BЈ
C C C C 100Њ LЈ LЈ OЈ O O OЈ O O OЈ
L B L B L B L B
hsm11gmse_0903_t08071.aihsm11gmse_0903_t08072.aihsm11gmse_0903_t08073.aihsm11gmse_0903_t08074.aiPearsonTEXAS.com 333 Problem 2 TEKS Process Standard (1)(F) Drawing Rotations in a Coordinate Plane PQRS has vertices P(1, 1), Q(3, 3), R(4, 1), and S(3, 0). What is the graph of r(90 , O)(PQRS)?
RЈ (Ϫ1, 4) y 4 SЈ (0, 3) QЈ (Ϫ3, 3) Q P R PЈ (Ϫ1, 1) x Ϫ6 Ϫ4 Ϫ2 O 2S 4 6 Ϫ2
How do you know where to draw the Find and graph the image of each vertex. Use the coordinate rule that describes a 90° vertices on the rotation about the origin: r(90°, O)(x, y) = (-y, x). coordinate plane? Use the rules for rotating P′ = r(90°, O)(1, 1) = (-1, 1) a point and apply them Q′ = r(90 , O)(3, 3) = (-3, 3) to each vertex of the ° geom12_se_ccs_c09l03_t04.ai figure. Then graph the R′ = r(90°, O)(4, 1) = (-1, 4) points and connect them to draw the image. S′ = r(90°, O)(3, 0) = (0, 3) Next, connect the vertices to graph P′Q′R′S′.
Problem 3
Using Properties of Rotations In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of the opposite sides of the parallelogram are equal? W Z What do you know about rotations that can help you show T that opposite sides of the parallelogram X Y are equal? You know that rotations Because T is the midpoint of the diagonals, XT = ZT and WT = YT. are rigid transformations, Since W and Y are equidistant from T, and the measure of ∠WTY = 180, so if you show that the you know that r (W) = Y . Similarly, r (X) = Z. opposite sides can be (180°, T) (180°, T) mapped to each other, You can rotate every point on WX in this same way, so r(180°, T)(WX) = YZ. then the side lengths Likewise, you can map WZ to YX with r(180°, T)(WZ) = YX. must be equal. geom12_se_ccs_c09l03_t05.ai Because rotations are rigid transformations and preserve distance, WX = YZ and WZ = YX.
334 Lesson 8-3 Rotations Problem 4 Identifying a Sequence of Transformations You are rearranging the furniture in your living room. Identify a sequence of translations and rotations that will move the sofa from its N Could you do these location in the southwest corner of the floor plan to a new location in steps in a different the northeast corner, facing west. order? Yes. For instance, you Step 1 Rotate the sofa 180° about the point marked by the black dot. could translate the sofa Step 2 Translate the sofa north until it is against the northwest corner. before rotating it. Step 3 Translate the sofa east until it is against the northeast corner.
N
180º
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For Exercises 1 and 2, use the graph below.
y 4 For additional support when F (0, 3) J (3, 2) completing your homework, 2 (go to PearsonTEXAS.com. G (؊4, 1 x Ϫ6 Ϫ4 O 4 6
(Ϫ4 H (1, ؊4
1. Graph r(90°, O)(FGHJ). 2. Graph r(270°, O)(FGHJ). 3. The coordinates of △PRS are P(-3, 2), R(2, 5), and S(0, 0). Use a coordinate rule to find the coordinates of the vertices of r(270°, O)(△PRS). 4. Create Representations to Communicate Mathematical Ideas (1)(E) Draw △LMN with vertices L(2, -1), M(6, -2), and N(4, 2). Find the coordinates of the vertices after a 90° rotation about the origin and about each of the points L, M, and N. 5. Explain Mathematical Ideas (1)(G) If you are given a figure and a rotation image of the figure, how can you find the center and angle of rotation? geom12_se_ccs_c09l03_t07.ai
PearsonTEXAS.com 335 6. Display Mathematical Ideas (1)(G) The Millenium Wheel, also known as the London Eye, contains Car 3 32 observation cars. Determine the angle of rotation that will bring Car 3 to the position of Car 18. 7. Explain Mathematical Ideas (1)(G) For center of rotation P, does an x° rotation followed by a y° rotation give the same image as a y° rotation followed by an x° rotation? Explain. 8. Describe how a series of rotations can have the same effect as a 360° rotation about a point X. 9. Create Representations to Communicate Mathematical Ideas (1)(E) Graph A(5, 2). Car 18 Graph B, the image of A for a 90° rotation about the origin O. Graph C, the image of A for a 180° rotation about O. Graph D, the image of A for a 270° rotation about O. What type of quadrilateral is ABCD? Explain. Point O is the center of the regular nonagon shown at the right. A B C 10. Analyze Mathematical Relationships (1)(F) Describe a rotation that I maps H to C. O D 11. Evaluate Reasonableness (1)(B) Your friend says that AB is the image of H ED for a 120° rotation about O. What is wrong with your friend’s statement? G E F Copy each figure and point P. Draw the image of each figure for the given rotation about P. Use prime notation to label the vertices of the image. 12. 60° 13. 90° 14. 180° P R E B hsm11gmse_0903_t09412.ai A D P
P D B T C R
15. In the diagram at the right, the figures are congruent. Identify a Figure 1 sequence of transformations that will carry Figure 1 to Figure 2. hsm11gmse_0903_t06754.ai 16. V′W′X′Y′ has vertices V′(-3,hsm11gmse_0903_t06755.ai 2), W′(5, 1), X′(0, 4), and Yhsm11gmse_0903_t06757.ai′(-2, 0). If r(90°, O)(VWXY) = V′W′X′Y′, what are the coordinates of VWXY? 17. A Ferris wheel is drawn on a coordinate plane so that the first car is located at the point (30, 0). What are the coordinates of the first car Figure 2 after a rotation of 270° about the origin?
336 Lesson 8-3 Rotations Connect Mathematical Ideas (1)(F) Use the diagram at the right. TQNV is a T Q rectangle. M is the midpoint of the diagonals. M 18. Can you use the properties of rotations to show that the lengths of the V N diagonals are equal? Explain. 19. Can you use properties of rotations to conclude that the diagonals of TQNV bisect the angles of TQNV? Explain.
20. Apply Mathematics (1)(A) Symbols are used in dictionaries to help users pronounce words correctly. The symbol is called a schwa. It is used in dictionariesgeom12_se_ccs_c09l03_t08.ai to represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What transformation maps a to a lowercase e? 21. A classmate says that the puzzle piece shown can fit into both Location A and Location B using only a sequence of translations and rotations. Is the classmate correct? Explain your reasoning by identifying a sequence A B of transformations that will carry the piece onto Locations A and B in the puzzle. 22. Use Representations to Communicate Mathematical Ideas (1)(E) Draw a bird’s-eye view of one room in your house, labeling the four cardinal directions (north, south, east, and west). Draw a second bird’s-eye view with one piece of furniture moved to a new location in the room. Identify a sequence of transformations that will carry the piece of furniture from its initial location to its new location.
TEXAS Test Practice
23. What is the image of (1, -6) after a 90° counterclockwise rotation about the origin? A. (6, 1) B. (-1, 6) C. (-6, -1) D. (-1, -6)
24. The costume crew for your school musical makes 18 in. aprons like the one shown. If blue ribbon costs 5 in. 5 in. $1.50 per foot, what is the cost of ribbon for six aprons? 5 in. 5 in. F. $15.75 H. $42.00 24 in. G. $31.50 J. $63.00
25. Use the following statement: If two lines are parallel, thenhsm11gmse_0903_t14047 the lines do not intersect. a. What are the converse, inverse, and contrapositive of the statement? b. What is the truth value of each statement you wrote in part (a)? If a statement is false, give a counterexample.
PearsonTEXAS.com 337 8-4 Symmetry
TEKS FOCUS VOCABULARY TEKS (3)(D) Identify and distinguish • Line of symmetry – See reflectional symmetry. between reflectional and rotational • Line symmetry – See reflectional symmetry. symmetry in a plane figure. • Point symmetry – A figure has point symmetry if it has 180° rotational TEKS (1)(C) Select tools, including symmetry. real objects, manipulatives, paper • Reflectional symmetry – A figure has reflectional symmetry, or line symmetry, and pencil, and technology as if there is a reflection for which the figure is its own image. The line of appropriate, and techniques, reflection is called the line of symmetry. It divides the figure into congruent including mental math, estimation, halves. and number sense as appropriate, to solve problems. • Rotational symmetry – A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is its own image. The angle of Additional TEKS (1)(D), (1)(E) rotation for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.
• Number sense – the understanding of what numbers mean and how they are related
ESSENTIAL UNDERSTANDING Some figures appear unchanged after a reflection across a line or a rotation about a point. Such figures are said to have symmetry .
Key Concept Types of Symmetry
A figure has line symmetry or reflectional symmetry if there is a reflection for which the figure is its own image. The line of reflection is called a line of symmetry. It divides the figure into congruent halves.
A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is its own image. The angle of rotation 1205 for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.
A figure with 180° rotational symmetry also has point symmetry. Each segment joining a point and its 180° rotation 1805 image passes through the center of rotation. A square, which has both 90° and 180° rotational symmetry, also has point symmetry.
338 Lesson 8-4 Symmetry Problem 1 TEKS Process Standard (1)(C) Identifying Lines of Symmetry How many lines of symmetry does a regular hexagon have? Select a tool (such as geoboards, pencil and paper, or geometry software) that will help you draw a diagram of a regular hexagon.
Use a pencil and paper to draw a diagram of a regular hexagon. Look for the ways the hexagon will reflect across a line onto itself.
The hexagon reflects onto itself across each line that passes through the midpoints of a pair of parallel sides.
The hexagon also reflects onto itself across each diagonal that passes through the center of the hexagon.
Count the lines of symmetry. A regular hexagon has six lines of symmetry.
Problem 2 Identifying Rotational Symmetry Does the figure appear to have rotational symmetry? If so, what is the angle of rotation? How do you identify rotational symmetry? A There is no center point about which the Look for a possible center triangle will rotate onto itself. This figure point. Think about the does not have rotational symmetry. angles formed by joining preimage-image pairs to the center. All these B The star has rotational angles must be congruent 725 for the figure to have symmetry. The angle of rotational symmetry. rotation is 72°.
PearsonTEXAS.com 339 Problem 3 TEKS Process Standard (1)(E)
Distinguishing Between Rotational and Reflectional Symmetry Does the plane figure appear to have rotational symmetry, reflectional symmetry, neither, or both? Explain your reasoning. How can you tell if a figure has rotational A B symmetry? A figure has rotational symmetry if there is a rotation of 180° or less for which the figure is unchanged. R
Both Neither A regular pentagon looks the same after being The letter R does not look the same after being rotated 72° about its center. So a regular rotated less than 180° about its center. So the pentagon has rotational symmetry. letter R does not have rotational symmetry. There are five lines shown that divide the There are no lines that divide the letter R in pentagon in half so that one half is the same half so that one half is the mirror image of the as the other. So a regular pentagon has other. So the letter R does not have reflectional reflectional symmetry. symmetry.
C D MM SS
Reflectional Symmetry Rotational Symmetry The letter M does not look the same after The letter S looks the same after being rotated being rotated less than 180° about its center. 180° about its center. So the letter S has So the letter M does not have rotational rotational symmetry. symmetry. There are no lines that divide the letter S in There is one line that divides the letter M half so that one half is the mirror image of the in half so that one half is the mirror image other. So the letter S does not have reflectional of the other. So the letter M has reflectional symmetry. symmetry.
340 Lesson 8-4 Symmetry NLINE O
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1. Display Mathematical Ideas (1)(G) Use the letters of the alphabets below. English: ABCDEFGHIJKLMNOPQRSTUVWXYZ
For additional support when Greek: ABGDEZHQIKLMNJOPRSTYFXCV completing your homework, Alphabet Symmetry go to PearsonTEXAS.com. Type of Symmetry Horizontal Vertical Language Line Line Point English Greek
a. Copy the table. Classify the letters of the alphabets. You will list some letters in more than one category. b. Which alphabet has more symmetrical letters? Explain. Identify whether each figure appears to have rotational symmetry, reflectional symmetry, neither, or both. If it has reflectional symmetry, sketch the figure and the line(s) of symmetry. If it has rotational symmetry, tell the angle of rotation. 2. 3. 4.
5. 6. 7.
8. 9. 10.
11. 12. 13.
PearsonTEXAS.com 341 Select Tools to Solve Problems (1)(C) Determine how many lines of symmetry each type of quadrilateral has. Select a tool, such as a geoboard or pencil and paper, to help you solve the problem. Include a sketch to support your answer. 14. rhombus 15. kite 16. square 17. parallelogram
18. If you stack the letters of MATH vertically, you can find a vertical line of symmetry. Find two other words for which this is true. 19. Connect Mathematical Ideas (1)(F) A quadrilateral with vertices at (1, 5) and (22, 23) has point symmetry about the origin. Show that the quadrilateral is a parallelogram. Tell what type(s) of symmetry each figure appears to have. For reflectional symmetry, sketch the figure and the line(s) of symmetry. For rotational symmetry, tell the angle of rotation. 20. 21.
22. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an angle also a line of symmetry of the angle? Explain. 23. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an angle of a triangle also a line of symmetry of the triangle? Explain. 10 83 24. The equation 10 - 1 = 0 , 83 is not only true, but also symmetrical (horizontally). Write four other equations or inequalities that are both true and symmetrical. Analyze Mathematical Relationships (1)(F) A figure that has a vertex at (3, 4) has the given line of symmetry. Tell the coordinates of another vertex of the figure. 25. the y-axis 26. the x-axis 27. the line y = x Use Representations to Communicate Mathematical Ideas (1)(E) Graph each equation and describe its symmetry.
28. y = x2 29. y = (x + 2)2 30. y = x3 31. y = |x|
342 Lesson 8-4 Symmetry For each three-dimensional figure, draw a net that has point symmetry and a net that has 1, 2, or 4 lines of symmetry. (A net is a two-dimensional diagram that you can fold to form a three-dimensional figure.) 32. 33.
Square pyramid
34. Do all regular polygons have rotational symmetry? Explain your reasoning. 35. Do all regular polygons have point symmetry? Explain your reasoning. 36. Use a straightedge to copy the rhombus at the right. a. How many lines of symmetry does the rhombus have? b. Draw all the lines of symmetry. 37. Do all parallelograms have reflectional symmetry? Explain your reasoning. Apply Mathematics (1)(A) Describe the types of symmetry, if any, of each logo. 38. 39. geom12_se_ccs_c09l03_t09.ai
40. 41.
TEXAS Test Practice
42. What is the smallest angle, in degrees, through which you can rotate a regular hexagon onto itself? 43. You place a sprinkler so that it is equidistant from three rose bushes B at points A, B, and C. How many feet is the sprinkler from A? 3 yd 44. △STU has vertices S(1, 2), T(0, 5), and U(28, 0). What is the 4 yd x-coordinate of S after a 270° rotation about the origin? A 45. The diagonals of rectangle PQRS intersect at O. PO = 2x - 5 and C OR = 7 - x. What is the length of QS?
PearsonTEXAS.com 343 8-5 Compositions of Rigid Transformations
TEKS FOCUS VOCABULARY TEKS (3)(A) Describe and perform • Glide reflection – the composition of a translation (a glide) and a transformations of figures in a plane using reflection across a line parallel to the direction of translation coordinate notation. Justify – explain with logical reasoning. You can justify a mathematical TEKS (1)(G) Display, explain, and justify • argument. mathematical ideas and arguments using precise mathematical language in written or • Argument – a set of statements put forth to show the truth or oral communication. falsehood of a mathematical claim
Additional TEKS (1)(D), (1)(E), (3)(B)
ESSENTIAL UNDERSTANDING You can express all rigid transformations as compositions of reflections.
Theorem 8-1
The composition of two or more rigid transformations is a rigid transformation.
Key Concept Classification of Rigid Transformations
There are only four kinds of rigid transformations.
Translation Rotation Reflection Glide Reflection
R
R R R R R R R R
OrientationsOrientations are are thethe same. OrientationsOrientations areare opposite.
Theorem 8-2 Reflections Across Parallel Lines hsm11gmse_0906_t09573.ai A composition of reflections across two parallel lines is AB a translation. BЈ You can write this composition as C (Rm ∘ R/)(△ABC) = △A″B″C″ AЈ CЈ AЉ Љ or Rm(R/(△ABC)) = △A″B″C″. B ᐍ AA″, BB″, and CC″ are all perpendicular to lines / and m. m CЉ
344 Lesson 8-5 Compositions of Rigid Transformations geom12_se_ccs_c09l04_t01.ai Theorem 8-3 Reflections Across Intersecting Lines
A composition of reflections across two intersecting lines is a rotation. m CЈ BЈ ᐉ You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″ CЉ B AЈ or Rm(R/(△ABC)) = △A″B″C″. A The figure is rotated about the point where the two lines AЉ Q intersect—in this case, point Q. BЉ C
Problem 1 geom12_se_ccs_c09l04_t02.ai Composing Reflections Across Parallel Lines
What is (Rm ∘ RO)( J)? What is the distance of the resulting translation? m ᐍ J
As you do the two reflections, keep track of the distance moved by a point P of the preimage.
How do you know that PA AP, Step 1 Reflect J across ᐍ. PA ϭ APЈ, so PPЈ ϭ 2APЈ. PB BP, and < > geom12_se_ccs_c09l04_t06.ai AB # O? m All three statements are ᐍ true by the definition of Step 2 Reflect the image across m. reflection across a line. Ј ϭ Љ Ј Љ ϭ Ј J P B BP , so P P 2P B. P A J PЈ B J PЉ P moved a total distance of 2APЈ ϩ 2PЈB, or 2AB.
The red arrow shows the translation. The total distance P moved is 2 AB. Because < > # AB # /, AB is the distance between / and m. The distance of the translation is twice the distance between / and m. geom12_se_ccs_c09l04_t07.ai
PearsonTEXAS.com 345 Problem 2 TEKS Process Standard (1)(G) Composing Reflections Across Intersecting Lines
Lines O and m intersect at point C and form a 70 angle. ᐍ m What is (Rm ∘ RO)(J)? What are the center of rotation and the angle of rotation for the resulting rotation? J 70Њ After you do the reflections, follow the path of a point P of the preimage. C Step 1 Reflect J across ᐍ.
J
ᐍ m PЈ J How do you show Step 2 Reflect the image across m. 3 that mj1 mj2? 2 4 geom12_se_ccs_c09l04_t08.ai J 1 If you draw PP and P PЉ label its intersection C point with line / as A, Step 3 Draw the angles formed by joining P, PЈ, and PЉ to C. then PA = P′A and PP′ # /. So, by the Converse of the Angle J is rotated clockwise about the intersection point of the lines. The center of rotation Bisector Theorem, is C. You know that m∠2 + m∠3 = 70. You can use the definition of reflection to m∠1 m∠2. = showgeom12_se_ccs_c09l04_t09.ai that m∠1 = m∠2 and m∠3 = m∠4. So m∠1 + m∠2 + m∠3 + m∠4 = 140. The angle of rotation is 140° clockwise.
Problem 3
Finding a Glide Reflection Image
Coordinate Geometry What is (Rx 0 ∘ T60, 57)(△TEX)? E y T 2 X x Ϫ4 Ϫ2 O 1
• The vertices of △TEX The image of First use the translation rule to • The translation rule △TEX for the glide translate △TEX. Then reflect the reflection translation image of each vertex across • The line of reflection hsm11gmse_0906_t09571.ai the line of reflection.
E y T 2 x Use the translation rule X Ϫ4 O 2 4 Reflect the image of ᭝TEX T (᭝TEX) to move <0, Ϫ5> EЈ across the line x ϭ 0. ᭝TEX down 5 units. TЈ XЈ
geom12_se_ccs_c09l04_t011.ai 346 Lesson 8-5 Compositions of Rigid Transformations Problem 4 TEKS Process Standard (1)(D) Determining Preimages Under Rigid Transformations
If (Rx@axis ∘ T<5, 0>) (△ABC) = △A″B″C″, then what are the coordinates of A, B, and C? Graph △ABC, △A′B′C′, and△A″B″C″.
The coordinates of △A″B″C″ are A″(3, -1), B″(4, -4), and y x C″(1, -2). △A″B″C″ is a transformation of △ABC, where O 2A 4 △ABC was translated 5 units right and then reflected across 2 - C the x-axis. You can determine the graph of △ABC by performing the transformations in reverse. -4 B Step 1 Reflect △A″B″C″ across the x-axis to find the vertices of △A′B′C′, the pre-image of △A″B″C″ before the B y B 4 How can you find the reflection. preimage of a figure C C 2 that was translated The vertices of △A′B′C′ are A′(3, 1), B′(4, 4), and 5 units to the right? C′(1, 2). A A x You can perform the -4 -2 O A 42 translation in reverse by Step 2 Translate △A′B′C′ 5 units left to find the vertices of 2 translating the figure △ABC, the pre-image of △A′B′C′ before the - C 5 units to the left. translation 5 units right. -4 B The vertices of △ABC are A(-2, 1), B(-1, 4), and C(-4, 2).
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Identify each mapping as a translation, a reflection, a rotation, or a glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection. For additional support when completing your homework, D G y K N go to PearsonTEXAS.com. 4
2 A F H J M P C E O 1 7 x Ϫ2
Ϫ4 B I L Q
1. △ABC S △EDC 2. △MNP S △EDC 3. △EDC S △PQM 4. △JLM S △MNJ 5. △PQM S △KJN 6. △HGF S △KJN 7. △ROS was reflected across the y-axis, then reflected across the x-axis, and then translated 2 unitshsm11gmse_0906_t09423.ai right. The resulting triangle has vertices at R‴(8, -3), O‴(4, 5), and S‴(-3, -6). What are the coordinates of R, O, and S?
PearsonTEXAS.com 347 Display Mathematical Ideas (1)(G) Find the image of each letter after the
transformation Rm ∘ RO. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation. 8. M ᐍ 9. m
m T ᐍ
10. 11. m 85Њ ᐍ hsm11gmse_0906_t06813.ai hsm11gmse_0906_t06812.aiC 75Њ N C
ᐍ L m
Graph △PNB and its image after the given transformation. y P 12. (R ∘ T )(△PNB) 2 hsm11gmse_0906_t06814.aiy = 3 62, 07 hsm11gmse_0906_t06816.ai x 13. (R ∘ T )(△PNB) O x = 0 60, -37 Ϫ2 2 14. (R ∘ T )(△PNB) N y = x 6-1, 17 B
15. Analyze Mathematical Relationships (1)(F) Let A′ be the point (1, 5). If (Ry = 1 ∘ T63, 07)(A) = A′, then what are the coordinates of A? hsm11gmse_0906_t06817.ai Create Representations to Communicate Mathematical Ideas (1)(E) Graph the preimage of each triangle in the coordinate plane before the given composition of transformations.
16. (Rx = 0 ∘ T<0, 4>)(△DEF) = △D″E″F″ 17. (Ry = 0 ∘ r<180°, O>)(△PQR) = △P″Q″R″ y y E 4 4
D 2 2 F x Q x -4 -2 O 42 -4 -2 O 42 -2 P -2
-4 -4 R
348 Lesson 8-5 Compositions of Rigid Transformations Describe the rigid transformation that maps the black figure onto the blue figure.
18. y 19. y 4 3 2 O x x O Ϫ3 Ϫ1 1 Ϫ4 Ϫ2 1 Ϫ2 Ϫ3
20. Describe a glide reflection that maps the black R to the blue R. Use the given points and lines. Graph AB and its image AB after a R
reflectionhsm11gmse_0906_t06818.ai first across O1 and then across O2. Is thehsm11gmse_0906_t06819.ai resulting transformation a translation or a rotation? For a translation, describe the direction and R distance. For a rotation, tell the center of rotation and the angle of rotation.
21. A(2, 4) and B(3, 1); /1: x-axis; /2: y-axis
22. A( -4, -3) and B( -4, 0); /1: y = x; /2: y = -x 23. A(6, -4) and B(5, 0); /1: x = 6; /2: x = 4 hsm11gmse_0906_t09422.ai 24. Connect Mathematical Ideas (1)(F) Does an x° rotation about a point P followed by a reflection across a line / give the same image as a reflection across / followed by an x° rotation about P? Explain.
TEXAS Test Practice
25. What is (Rx = 0 ∘ T<-12, -6>)(11, -5)? A. (1, -11) B. (-1, 11) C. (1, 11) D. (-1, -11) 26. ABCD is a rectangular window divided into 12 panes of glass. E, F, G, and A E B H are midpoints of AB, BC, CD, and AD, respectively. Which statement must be true? F. The quadrilateral panes are squares. H F G. The quadrilateral panes are rhombuses. H. The triangular panes are all congruent. D G C J. The triangular panes are right triangles. 27. A triangle has side lengths 7 in., 9 in., and x in. Which inequality must be true? A. 7 6 x 6 9 B. -2 6 x 6 9 C. 2 6 x 6 16 D. 7 6 x 6 16 hsm11gmse_0906_t14039 28. △ABC and △HIG are acute triangles such that △ABC ≅ △HIG. BL and IT are altitudes of the two triangles. Is BL ≅ IT? Justify your answer.
PearsonTEXAS.com 349 8-6 Congruence Transformations
TEKS FOCUS VOCABULARY TEKS (6)(C) Apply the definition of • Congruent – Two figures are congruent if and only if there is a congruence, in terms of rigid transformations, sequence of one or more rigid transformations that maps one figure to identify congruent figures and their onto the other. corresponding sides and angles. • Congruence transformation – a transformation in which an original TEKS (1)(F) Analyze mathematical figure and its image are congruent relationships to connect and communicate mathematical ideas. • Analyze – closely examine objects, ideas, or relationships to learn more about their nature Additional TEKS (1)(D), (1)(G), (3)(A), (3)(B), (3)(C)
ESSENTIAL UNDERSTANDING You can use compositions of rigid transformations to understand congruence.
Key Concept Congruent Figures
Two figures are congruent if and only if there is a sequence of one or more rigid transformations that maps one figure onto the other. This is a second way to define congruence.
Problem 1 TEKS Process Standard (1)(F) Identifying Corresponding Sides and Angles
The composition (Rn ∘ r(90°, P))(LMNO) ∙ GHJK is shown at the right. Since LMNO maps to GHJK by a sequence of L M rigid transformations, the figures are congruent. How can you use the properties of rigid A Which angle pairs have equal measures? transformations to O N G P find equal angle Because compositions of rigid transformations preserve measures and equal angle measure, corresponding angles have equal measures. H K side lengths? n Rigid transformations m∠L = m∠G, m∠M = m∠H, m∠N = m∠J, and J preserve angle measure m∠O = m∠K and distance, so identify corresponding angles B Which sides have equal lengths? and corresponding side By definition, rigid transformations preserve distance. So lengths. geom12_se_ccs_c09l05_t02.ai corresponding side lengths have equal measures.
LM = GH, MN = HJ, NO = JK, and LO = GK
350 Lesson 8-6 Congruence Transformations Problem 2 Identifying Congruent Figures Which pairs of figures in the grid are congruent? y 6 For each pair, what is a sequence of rigid E Q transformations that maps one figure to the other? 4 Does one rigid P X W transformation count Figures are congruent if and only if there is a sequence 2 as a sequence? of rigid transformations that maps one figure to the Yes. It is a sequence of other. So, to find congruent figures, look for sequences D F Y Z x length 1. O of translations, rotations, and reflections that map one Ϫ6 Ϫ4 Ϫ2 2 4 6 AB L figure to another. Ϫ2 N
Because r (△DEF) = △LMN, the triangles are (180°, O) J C congruent. Because (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ, G Ϫ6 M the trapezoids are congruent. Because T6-2, 97(HG) = PQ, H the line segments are congruent.
Problem 3
Identifying Congruence Transformations
In the diagram at the right, △JQV @ △EWT. What is a y E congruence transformation that maps △JQV onto △EWT ? geom12_se_ccs_c09l05_t03_updated.ai4
T 2 The coordinates of the vertices of A sequence of rigid transformations the triangles that maps △JQV onto △EWT W x Ϫ4 Ϫ2 O J 4 Ϫ2 Identify the corresponding parts and find a congruence transformation Ϫ4 V that maps the preimage to the image. Then use the vertices to verify the Q congruence transformation.
Because △EWT lies above △JQV on the plane, a translation y E can map △JQV up on the plane. Also, notice that △EWT is on 4 the opposite side of the y-axis and has the opposite orientation of △JQV. This suggests that the triangle is reflected across the T 2 y-axis. W x It appears that a translation of △JQV up 5 units followed by 4 2 O J 4 a reflection across the y-axis maps △JQV to △EWT. Verify by geom12_se_ccs_c09l05_t05.ai2 using the coordinates of the vertices. 4 V Q T60, 57(x, y) = (x, y + 5)
T60, 57(J) = (2, 4)
Ry@axis(2, 4) = (-2, 4) = E
continued on next page ▶
PearsonTEXAS.com 351
geom12_se_ccs_c09l05_t06.ai Problem 3 continued Next, verify that the sequence maps Q to W and V to T.
T60, 57(Q) = (1, 1) T60, 57(V) = (5, 2)
Ry@axis(1, 1) = (-1, 1) = W Ry@axis(5, 2) = (-5, 2) = T
So the congruence transformation Ry@axis ∘ T60, 57 maps △JQV onto △EWT. Note that there are other possible congruence transformations that map △JQV onto △EWT.
Problem 4 TEKS Process Standard (1)(G)
Proof Verifying the SAS Postulate J Given: ∠J ≅ ∠P, PA ≅ JO, FP ≅ SJ How do you show F A O that the two triangles Prove: △JOS ≅ △PAF are congruent? S Find a congruence transformation that maps P one onto the other. Step 1 Translate △PAF so that points A and O coincide. J
F O A geom12_se_ccs_c09l05_t08.ai S P Step 2 Because PA ≅ JO, you can rotate △PAF about point A so that PA and JO coincide. F J P geom12_se_ccs_c09l05_t09.ai A O S Step 3 Reflect △PAF across PA. Because reflections preserve angle measure and distance, and because ∠J ≅ ∠P and FP ≅ SJ, you know that the reflection maps ∠P to ∠J and FP to SJ. Since points S and F coincide, △PAF coincides with △JOS. geom12_se_ccs_c09l05_t010.aiJ P
A S O F There is a congruence transformation that maps △PAF onto △JOS, so △PAF ≅ △JOS.
352 Lesson 8-6 Congruence Transformations geom12_se_ccs_c09l05_t011.ai NLINE O
H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O
For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image.
For additional support when 1. y 2. y 3. y 4 4 4 completing your homework, V E C F go to PearsonTEXAS.com. J A 2 D G x x K I x Ϫ4 Q 2 4 Ϫ4 Ϫ2 O 4 Ϫ4 E D F B Y T Ϫ2 T L R M A GϪ4 Ϫ4 Ϫ4 C B S
4. Apply Mathematics (1)(A) Artists frequently use congruence transformations in their work. The artworks shown below are called tessellations. What types of congruence transformations can you identify in the tessellations? a. b.
geom12_se_ccs_c09l05_t016.ai geom12_se_ccs_c09l05_t017.ai geom12_se_ccs_c09l05_t018.ai
Analyze Mathematical Relationships (1)(F) Find a congruence transformation that maps △LMN to △RST.
5. y 6. y M 4 4 L L T S 2 2 N x x O 2 -4 -2 O 2 4 T MN Ϫ4 R R S
7. Verify the ASA Postulate for triangle congruence by using E K Proof congruence transformations. L Given: EK ≅ LH Prove: △EKS ≅ △HLA ∠ E ≅ ∠H S A ∠ K ≅ ∠L H
geom12_se_ccs_c09l05_t019.ai PearsonTEXAS.com 353
geom12_se_ccs_c09l05_t022.ai 8. Justify Mathematical Arguments (1)(G) Verify the AAS Postulate N Proof for triangle congruence by using congruence transformations. Given: ∠ I ≅ ∠V Prove: △NVZ ≅ △CIQ V Z ∠ C ≅ ∠N I Q QC ≅ NZ C 9. If two figures are ______, then there is a sequence of rigid transformation that maps one figure onto the other. 10. The graph at the right shows two congruent isosceles triangles. y What are four different rigid transformations that map the top triangle onto the bottom triangle?
11. Prove the statements in parts (a) and (b) to show that geom12_se_ccs_c09l05_t023.aix Proof congruence in terms of transformations is equivalent to the Ϫ4 Ϫ2 O 2 4 criteria for triangle congruence you learned in Topic 4. a. If there is a congruence transformation that maps △ABC to △DEF, then corresponding pairs of sides and corresponding pairs of angles are congruent. b. In △ABC and △DEF, if corresponding pairs of sides and corresponding pairs of angles are congruent, then there is a congruence transformation that maps △ABC to △DEF. 12. Apply Mathematics (1)(A) Cookie makers often use cookie cutters so that the cookies all look the same. The baker fills a cookie sheet for baking as shown. What types of congruence transformations can you use to geom12_se_ccs_c09l05_t032.ai show that the cookies are congruent to one another? 13. Use congruence transformations to prove the Proof Isosceles Triangle Theorem. H Given: FG ≅ FH Prove: ∠G ≅ ∠H F
G
TEXAS Test Practice geom12_se_ccs_c09l05_t031.ai
14. In △FGH and △XYZ, ∠G and ∠Y are right angles. FH ≅ XZ and GH ≅ YZ. If GH = 7 ft and XY = 9 ft, what is the area of △FGH in square inches? 15. A classmate says that a certain regular polygon has 50° rotational symmetry. Explain your classmate’s error.
354 Lesson 8-6 Congruence Transformations Activity Lab Exploring Dilations
Use With Lesson 8-7 teks (3)(A), (1)(E)
In this activity, you will explore the properties of dilations. A dilation is defined by a center of dilation and a scale factor. 1
To dilate a segment by a scale factor n with center of dilation at the origin, you y GЈ measure the distance from the origin to each point on the segment. The diagram 8 at the right shows the dilation of GH by the scale factor 3 with center of dilation at the origin. To locate the dilation image of GH, draw rays from the origin through 6 points G and H. Then, measure the distance from the origin to G. Next, find the point along the same ray that is 3 times that distance. Label the point G′. Now 4 G dilate the endpoint H similarly. Draw G H . ′ ′ 2 1. Graph RS with R(1, 4) and S(2, -1). What is the length of RS? x O 4 6 2. Graph the dilations of the endpoint of RS by scale factor 2 and center of dilation Ϫ2 H at the origin. Label the dilated endpoints R′ and S′. HЈ 3. What are the coordinates of R′ and S′? 4. Graph R′S′. 5. What is R′S′? 6. How do the lengths of RS and R′S′ compare? 1 7. Graph the dilation of RS by scale factor 2 with center of dilation at the origin. Label the dilation R S . ″ ″ geo12_se_ccs_c09l06a_patches.ai 8. What is R″S″? 9. How do the lengths of R′S′ and R″S″ compare? 10. What can you conjecture about the length of a line segment that has been dilated by scale factor n? 2
11. Draw a line on a coordinate grid that does not pass through the origin. Use the method in Activity 1 to construct several dilations of the line you drew with different scale factors (not equal to 1). Make a conjecture relating the slopes of the original line and the dilations. 12. On a new coordinate grid, draw a line through the origin. What happens when you try to construct a dilation of this line? Explain.
PearsonTEXAS.com 355 8-7 Dilations
TEKS FOCUS VOCABULARY TEKS (3)(A) Describe and • Center of dilation – See dilation. • Reduction – A dilation is a reduction if the perform transformations • Dilation – A dilation with center of scale factor n is between 0 and 1. of figures in a plane using dilation C and scale factor n, where n 7 0, • Scale factor of a dilation – the ratio of the coordinate notation. is a transformation that maps a point R distances from the center of dilation to an > TEKS (1)(D) Communicate to R′ in such a way that R′ is on CR and image point and to its preimage point. mathematical ideas, CR′ = n # CR. The image of C is itself. reasoning, and their • Enlargement – A dilation is an • Implication – a conclusion that follows implications using multiple enlargement if the scale factor n is from previously stated ideas or reasoning representations, including greater than 1. without being explicitly stated symbols, diagrams, • Non-rigid transformation – a • Representation – a way to display or graphs, and language as transformation in the plane that does describe information. You can use a appropriate. not necessarily preserve distance or angle representation to present mathematical Additional TEKS (1)(F), measure ideas and data. (1)(G), (3)(B) • Ratio – A ratio is a comparison of two quantities by division. You can write the ratio of two numbers a and b, where a b ≠ 0, in three ways: , a : b, or a to b. b
ESSENTIAL UNDERSTANDING You can use a scale factor to make a larger or smaller copy of a figure.
Key Concept Dilation
A dilation with center of dilation C and scale factor n, n 7 0, can be written as D(n, C). A dilation is a transformation with the following properties. PЈ
P Q QЈ
CЈ ϭ C R RЈ
CRЈ ϭ n и CR S • The image of C is itself (that is, C′ = C). > CR′ • For any other point R, R′ is on CR and CR′ = n # CR, or n = CR . • Dilations preservehsm11gmse_0905_t09541.ai angle measure.
356 Lesson 8-7 Dilations Key Concept Dilations Centered at the Origin
For a dilation centered at the origin, you can find the image of y (a point P(x, y) by multiplying the coordinates of P by the scale ny P(nx, ny factor n. The coordinate rule for a dilation of scale factor n with center of dilation at the origin can be written as shown below. y P(x, y) n ؒ OP ؍ Dn (x, y) = (nx, ny) OP x O x nx
Key Concept Dilations Not Centered at the Origin
The center of a dilation can be any point C(h, k) y hsm11gmse_0905_t08148.ai in the plane. Using a composition of a translation, P (n(x h) h, n(y k) k) a dilation centered at the origin, and a second Q translation, you can write the following coordinate P(x, y) rule for D(n, C). Q D(n, C)(x, y) = (n(x - h) + h, n(y - k) + k) C(h, k) R R x O
Step 1 Use the translation T6-h, -k7 to move the (x, y) S (x - h, y - k) center of dilation to the origin.
Step 2 Use the dilation Dn to dilate by scale (x - h, y - k) S (n(x - h), n(y - k)) factor n.
Step 3 Use the translation T6h, k7 to move the (n(x - h), n(y - k)) S (n(x - h) + h, n(y - k) + k) center of dilation back to (h, k).
Problem 1 Finding a Scale Factor
Multiple Choice Is D(n, X)(△XTR) △X′T′R′ an enlargement RЈ TЈ or a reduction? What is the scale factor n of the dilation? 1 8 Why is the scale enlargement; n = 2 reduction; n = 3 R 4 1 T factor not 12, or 3? enlargement; n = 3 reduction; n = 3 4 The scale factor of a The image is larger than the preimage, so the dilation is an XЈ ϭ X dilation always has the image length (or the enlargement. distance between a point Use the ratio of the lengths of corresponding sides to find the scale factor.
on the image and the n X′T′ 4 + 8 12 3 center of dilation) in the = XT = 4 = 4 = hsm11gmse_0905_t08144.ai numerator. △X ′T ′R′ is an enlargement of △XTR, with a scale factor of 3. The correct answer is B.
PearsonTEXAS.com 357 Problem 2 TEKS Process Standard (1)(G) Finding a Dilation Image What are the coordinates of the vertices of y Will the vertices of Z x the triangle move D2(△PZG)? Graph the image of △PZG. closer to (0, 0) or Ϫ4 Ϫ2 O 4 P farther from (0, 0)? Identify the coordinates of each vertex. The center of The scale factor is 2, dilation is the origin and the scale factor is 2, so use the G so the dilation is an coordinate rule D2(x, y) = (2x, 2y). enlargement. The vertices y will move farther from D2(P) = (2 # 2, 2 # (-1)), or P′(4, -2). Z Z 2 (0, 0). x D2(Z) = (2 # (-2), 2 # 1), or Z′(-4, 2). geom12_se_ccs_c09l06_t03.ai−4 O P 4 D2(G) = (2 # 0, 2 # (-2)), or G′(0, -4). G P To graph the image of △PZG, graph P′, Z′, and G′. Then draw △P′Z′G′. G
Problem 3
Composing Rigid Transformations, Including a Dilation How can a dilation be a rigid transformation? Determine the image of △FRM after a dilation centered y 4 If the scale factor of the at (0, 0) with scale factor 1, composed with a translation R dilation is 1, then the 4 units down. preimage and the image M are congruent. Find the coordinates of the vertices of F x △F′R′M′ and △F″R″M″. −4 −2 O 2
The coordinate rule that describes the dilation is (x, y) S (1x, 1y). Use the rule to find F′, R′, and M′.
F(-4, 0) S F′(-4, 0) R(1, 3) S R′(1, 3) M(3, 2) S M′(3, 2) The coordinate rule that describes the translation is (x, y) S (x, y - 4). Use the rule to find F″, R″, and M″. y F′( -4, 0) S F″( -4, -4) R = R M = M R′(1, 3) S R″(1, -1) F = F x M′(3, 2) S M″(3, -2) −2 O R Draw △FRM, △F′R′M′, and △F″R″M″ on the coordinate plane. M −4 F
358 Lesson 8-7 Dilations Problem 4 TEKS Process Standard (1)(D) Determining the Image of a Dilation Not Centered at the Origin Use ▱HJMN shown at the right.
A Write a coordinate rule that describes a dilation y 1 centered at J with scale factor 2. N x 4 2 2 4 You can use the composition of a translation, a − − O dilation, and a second translation to find a H −2 M coordinate rule for D 1 (x, y). (2, J) −4 J The coordinates of J are (-2, -4). Use these coordinates to identify the two translations you need What translation will to use in the composition. move the center of dilation J(–2, –4) to Step 1 Translate (x, y) 2 units right and 4 units up to move the the origin (0, 0)? center of dilation from J(-2, -4) to (0, 0). J must move 2 units right and 4 units up to be at (x, y) S (x + 2, y + 4) (0, 0). 1 Step 2 Dilate by a scale factor of 2. 1 1 1 1 (2(x + 2), 2(y + 4)) = (2x + 1, 2y + 2) Step 3 Translate 2 units left and 4 units down to move the center of dilation from (0, 0) back to J(-2, -4). 1 1 1 1 (2x + 1 - 2, 2y + 2 - 4) = (2x - 1, 2y - 2) The coordinate rule that describes the dilation is D 1 (x, y) 1x 1, 1y 2 . (2, J) = (2 - 2 - )
B Graph D 1 (HJMN). (2, J) You can find the coordinates of the vertices of H′J′M′N′ by applying the coordinate rule you wrote in Part A. 1 1 H(-4, -2) S (2(-4) - 1, 2(-2) - 2), or H′(-3, -3) 1 1 J(-2, -4) S (2(-2) - 1, 2(-4) - 2), or J′(-2, -4) 1 1 M(2, -2) S (2 # 2 - 1, 2(-2) - 2), or M′(0, -3) 1 1 N(0, 0) S (2 # 0 - 1, 2 # 0 - 2), or N′(-1, -2) Graph H′J′M′N′. y N x −4 −2 O 2 4 N H M H −4 M J = J
PearsonTEXAS.com 359 Problem 5
Determining the Image of a Composition of Rigid and Non-Rigid Transformations △ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of the image of △ABC after a dilation with scale factor 2 and center of dilation at point A, followed by a translation 5 units to the left. Graph the image. How do you find the rule for a dilation B0 y B9 centered at (22, 22)? 4 You can write the rule of a dilation not 2 centered at the origin B x using a composition of -8 -4 -2 O a translation, a dilation, A9 and a second translation A0 C0 A C C9 of coordinate (x, y).
Step 1 The coordinate rule that describes the dilation is (x, y) S (2x + 2, 2y + 2). A(-2, -2) S A′(-2, -2) B(0, 1) S B′(2, 4) C(0, -2) S C′(2, -2) Step 2 The coordinate rule that describes the translation is (x, y) S (x - 5, y). A′(-2, -2) S A″(-7, -2) B′(2, 4) S B″(-3, 4) C′(2, -2) S C″(-3, -2)
Problem 6 Using a Scale Factor to Find a Length
What does a scale Biology A magnifying glass shows you an image of an factor of 7 tell you? object that is 7 times the object’s actual size. So the scale A scale factor of 7 tells factor of the enlargement is 7. The photo shows an apple you that the ratio of seed under this magnifying glass. What is the actual length the image length to the of the apple seed? actual length is 7, or image length The enlarged length of the apple seed is 1.75 in. Set up an equation = 7. 1.75 in. actual length to find the actual length of the apple seed. 1.75 = 7 # p image length = scale factor # actual length 0.25 = p Divide each side by 7. The actual length of the apple seed is 0.25 in.
360 Lesson 8-7 Dilations NLINE O
H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O
The blue figure is a dilation image of the black figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation. For additional support when completing your homework, 1. 2. 3. go to PearsonTEXAS.com. 2
4 4 A 6 R L
4. 5. y 6. y hsm11gmse_0905_t06790.ai 6 6 hsm11gmse_0905_t06792.ai hsm11gmse_0905_t06794.ai 4 4
2 M x x O 2 4 6 Ϫ6 Ϫ4 Ϫ2 O Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Write a coordinate rule that describes each dilation. Use your rule to find the imageshsm11gmse_0905_t06795.ai of the vertices of △PQR for each dilation. Graph the image. hsm11gmse_0905_t06796.aihsm11gmse_0905_t06798.ai 7. D10 (△PQR) 8. D3 (△PQR) 9. D(3, Q) (△PQR) 4 Q y y y Q Q 4 2 P 1 2 O x x Ϫ4 1 P O 2 x O 51 P R Ϫ3 Ϫ3 Ϫ2 R R
Apply Mathematics (1)(A) You look at each object described in Exercises 10–12 under a magnifying glass. Find the actual dimension of each object.
10. Thehsm11gmse_0905_t06800.ai image of a button is 5 timeshsm11gmse_0905_t06801.ai the button’s actual size andhsm11gmse_0905_t06799.ai has a diameter of 6 cm. 11. The image of an ant is 7 times the ant’s actual size and has a length of 1.4 cm. 12. The image of a capital letter N is 6 times the letter’s actual size and has a height of 1.68 cm. Find the image of each point for the given dilation.
13. L(-3, 0); D5 (L) 14. N(-4, 7); D(0.2, N) (N) 15. A(-6, 2); D1.5 (A)
PearsonTEXAS.com 361 Use the graph at the right. Find the vertices of the image of QRTW for the y W given transformation or composition of transformations. Q 4 16. a dilation with scale factor 100 2 1 17. a dilation with scale factor 2 centered at (0, 2) followed by a T translation 3 units down O x Ϫ3 Ϫ1 2 18. D(1,O) ∘ r(180°,Q) R
19. D(10,T) ∘ Ry-axis
20. The vertices of △S″U″J″ are S″ (-1,-1), U″ (0, 1), and J″ (1,-1). Suppose the composition of transformations resulting in △S″U″J″ was a dilation with scale 1 hsm11gmse_0905_t06802.ai factor 3 centered at point S followed by a translation 4 units up. Determine the coordinates of △SUJ. Then, graph △SUJ. Display Mathematical Ideas (1)(G) Graph MNPQ and its image MNPQ for a dilation with center (0, 0) and the given scale factor.
21. M(1, 3), N(-3, 3), P(-5, -3), Q(-1, -3); 3 1 22. M(2, 6), N(-4, 10), P(-4, -8), Q(-2, -12); 4 23. Select Tools to Solve Problems (1)(C) Use the dilation command in geometry software or drawing software to create a design that involves repeated dilations, such as the one shown at the right. The software will prompt you to specify a center of dilation and a scale factor. Print your design and color it. Feel free to use other transformations along with dilations.
24. Let / be a line through the origin. Show that Dk(/) = / by showing that if C = (c1, c2) is on /, then Dk(C) is also on /. 25. Let A = (a , a ) and B = (b , b ), let A′ = D (A) and B′ = D (B) 1 2 1 <2 > k k with k ≠ 1, and suppose that AB does not pass through the origin. < > < > a. Show that AB and A′B′ are not the same line. < > < > b. Suppose that a1 ≠ b1. Show that AB is parallel to A′B′ by showing that they have the same slope. hsm11gmse_0905_t09542.ai < > < > c. Show that AB } A′B′ if a1 = b1. 26. Explain Mathematical Ideas (1)(G) You are given AB and its dilation image A′B′ with A, B, A′, and B′ noncollinear. Explain how to find the center of dilation and scale factor. 27. Explain Mathematical Ideas (1)(G) The diagram at the right shows △LMN and its image △L′M′N′ for a dilation with LЈ x ϩ 3 4 center P. Find the values of x and y. Explain your reasoning. L MЈ x 2 M P yЊ N NЈ (2y Ϫ 60)Њ
362 Lesson 8-7 Dilations hsm11gmse_0905_t06804.ai 28. Analyze Mathematical Relationships (1)(F) An equilateral triangle has 4-in. sides. Describe its image for a dilation with center at one of the triangle’s vertices and scale factor 2.5. In the coordinate plane, you can extend dilations to include scale factors that are negative numbers. For Exercises 29 and 30, use △PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).
29. Graph D-3 (△PQR).
30. a. Graph D-1 (△PQR). b. Explain why the dilation in part (a) may be called a reflection through a point. 31. Use Representations to Communicate Mathematical Ideas (1)(E) A flashlight projects an image of rectangle ABCD on a wall so that each vertex of ABCD is 3 ft away from the corresponding vertex of A′B′C′D′. The length of AB is 3 in. The length of A′B′ is 1 ft. How far from each vertex of ABCD is the light? B9 C9 B C 9 A D A
D9 32. Determine the image of △TRI after a rotation of 180° y 4 around T composed with a dilation with scale factor 1. R 33. Under a dilation, what scale factor will preserve congruence? I T x −4 −2 O 2
TEXAS Test Practice
34. A dilation maps △CDE onto △C′D′E′. If CD = 7.5 ft, CE = 15 ft, D′E′ = 3.25 ft, and C′D′ = 2.5 ft, what is DE? A. 1.08 ft B. 5 ft C. 9.75 ft D. 19 ft 35. You want to prove indirectly that the diagonals of a rectangle are congruent. As the first step of your proof, what should you assume? F. A quadrilateral is not a rectangle. G. The diagonals of a rectangle are not congruent. H. A quadrilateral has no diagonals. J. The diagonals of a rectangle are congruent. 36. Which word can describe a kite? A. equilateral B. equiangular C. convex D. scalene
PearsonTEXAS.com 363 8-8 Other Non-Rigid Transformations
TEKS FOCUS VOCABULARY TEKS (3)(A) Describe and perform transformations • Compression – a transformation that decreases the distance of figures in a plane using coordinate notation. between corresponding points of a figure and a line
TEKS (1)(E) Create and use representations to • Stretch – a transformation that increases the distance between organize, record, and communicate mathematical corresponding points of a figure and a line ideas. • Representation – a way to display or describe information. You Additional TEKS (1)(A), (1)(D), (1)(F), (3)(B), can use a representation to present mathematical ideas and data. (3)(C)
ESSENTIAL UNDERSTANDING You can change the size of a figure in the coordinate plane by multiplying the x- and y-coordinates by different factors. You can compose this type of transformation with the other transformations you have learned.
Key Concept Other Non-Rigid Transformations
A stretch is a transformation that increases the distance between corresponding points of a figure and a line. A compression is a transformation that decreases the distance between corresponding points of a figure and a line. Horizontal Stretch Vertical Stretch (x, y) S (ax, y), where a 7 1 (x, y) S (x, by), where b 7 1 y y y y
x x x x O O O O
Horizontal Compression Vertical Compression (x, y) S (ax, y), where 0 6 a 6 1 (x, y) S (x, by), where 0 6 b 6 1 y y y y
x x x x O O O O
364 Lesson 8-8 Other Non-Rigid Transformations Problem 1 TEKS Process Standard (1)(E) Performing a Stretch Quadrilateral EFGH has vertices E(22, 2), F(2, 2), G(2, 22), and H(22, 22). What are the coordinates of the vertices of the image of EFGH after the transformation How can you tell the (x, y) S (3x, 2y)? Graph the image of EFGH. difference between You can think of this transformation as a composition of a horizontal stretch and a a compression and a vertical stretch. The horizontal stretch factor is 3. The vertical stretch factor is 2. stretch? In a stretch, at least one Use the coordinate rule (x, y) S (3x, 2y) to find the y coordinate is multiplied coordinates of the images of the vertices. EЈ FЈ by a factor greater than 1. 4 In a compression, at E(-2, 2) S E′(-6, 4) E F least one coordinate is 2 multiplied by a factor less F(2, 2) S F′(6, 4) x than 1. G(2, -2) S G′(6, -4) Ϫ6 Ϫ4 Ϫ2 O 2 4 6 Ϫ2 H(-2, -2) S H′(-6, -4) H G Ϫ4 Graph EFGH and E′F′G′H′. HЈ GЈ
Problem 2 Describing a Non-Rigid Transformation
Is △STW S △S′T′W′ a vertical compression or a vertical stretch? T y Write a coordinate rule that maps △STW to △S′T′W′. 4
Since the image appears to be shorter than the preimage, this 2 transformation is a vertical compression. T9 O x Compare corresponding vertical distances for the figures to determine -4S9 -2 W9 the vertical compression factor that maps △STW to △S′T′W′. -2 Why aren’t T9W9 and TW used to find the S′T′ 2 1 b = = = -4 vertical compression ST 8 4 S W factor? 1 So △S′T′W′ is a vertical compression of △STW by a factor of 4. TW and T9W9 are not 1 The coordinate rule that describes the transformation is (x, y) S (x, y). vertical distances, so it 4 would be more difficult to use them.
PearsonTEXAS.com 365 Problem 3
Determining the Image of a Composition of Non-Rigid Transformations △ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of the image of △ABC after a dilation with scale factor 2 centered at the origin, followed by the horizontal stretch (x, y) S (2x, y). Graph the image. Step 1 The coordinate rule that describes the dilation is (x, y) S (2x, 2y). A(-2, -2) S A′(-4, -4) B(0, 1) S B′(0, 2) C(0, -2) S C′(0, -4) Does the stretch affect the points B Step 2 Apply the rule (x, y) S (2x, y) for the stretch to y 2 and C ? the image of the dilation. B¿ = B– No, because B′ and C′ B x A ( 4, 4) A ( 8, 4) are on the y-axis. ′ - - S ″ - - -8 -6 O 2 B′(0, 2) S B″(0, 2) A C C′(0, -4) S C″(0, - 4) C¿ = C– A– A¿
Problem 4 Determining the Preimage of a Composition of Non-Rigid Transformations
The vertices of △P″Q″R″ are P″(-1, -1), Q″(0, 1), and R″(1, -1). △PQR maps to 1 △P″Q″R″ through a dilation with scale factor 3 centered at (0, -1) followed by the How can you find compression (x, y) S 1 x, y . Determine the coordinates of the vertices of △PQR. the coordinates of (2 ) the vertices of the Then graph △PQR. preimage? First, reverse the transformation (x, y) 1 x, y by multiplying the x-coordinate of To find the coordinates S (2 ) for the preimage, multiply each vertex of △P″Q″R″ by 2. The vertices of △P′Q′R′ are P′(-2, -1), Q′(0, 1), and the x-coordinate of the R′(2, -1). image by the reciprocal 1 of the given horizontal Then reverse the dilation with scale factor 3 centered at Q y (0, 1) by translating to bring the center of dilation to compression factor. - 4 the origin, dilating with scale factor 3, and translating to bring the center of dilation back to (0, -1). Q = Q This sequence of transformations gives the coordinate x rule (x, y) S (6x, 3(y + 1) - 1). The vertices of △PQR Ϫ6 Ϫ4 Ϫ2 O 2 4 6 are P(-6, -1), Q(0, 5), and R(6, -1). P P P R R R
366 Lesson 8-8 Other Non-Rigid Transformations Problem 5 TEKS Process Standard (1)(A) Identifying a Sequence of Transformations An architect’s plan for a city park is shown on the coordinate grid at the left below. The mayor of the city asks that the swimming pool be 50, longer, but not wider, and wants to move it to the other end of the park. Describe a sequence of transformations that will move the pool to the outlined location on the architect’s plan. The pool is a rectangle with vertices A(1, 2), B(1, 6), C(4, 6), and D(4, 2). ABCD will need to be vertically stretched and then translated horizontally and vertically to be mapped to EFGH. How can you 3 determine the vertical Step 1 Stretch ABCD using the rule (x, y) S (x, 2 y) stretch factor? The vertical stretch factor A(1, 2) S A′(1, 3) is the ratio of the vertical B(1, 6) S B′(1, 9) length of the outlined location, which is 6 units, C(4, 6) S C′(4, 9) to the length of the D(4, 2) S D′(4, 3) original pool, which is
4 units. The vertical 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 stretch factor is 3. 0 2 FG 14 14
1 12 12
2 10 10
3 B9 C9 8 8 EH 4 BC 6 6
5 4 4
6 A9 D9 2 2 A D
7 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
Step 2 Rectangle EFGH appears to be translated 5 units up and 10 units to the right from A′B′C′D′. Find the coordinates using the rule (x, y) S (x + 10, y + 5). A′(1, 3) S E(11, 8) B′(1, 9) S F(11, 14) C′(4, 9) S G(14, 14) D′(4, 3) S H(14, 8) The pool can be moved to the outlined location through a stretch followed by a translation.
PearsonTEXAS.com 367 NLINE O
H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O
Find the vertices of each figure’s image after the given transformation. Then graph the image. 1 1 1 For additional support when 1. (x, y) S (2x, 4y) 2. (x, y) S (2x, 2 y) 3. (x, y) S (2 x, 3 y) completing your homework, y H y J y go to PearsonTEXAS.com. 2 Q R AB x 2 2 Ϫ2 O 2 x CD x O Ϫ2 Ϫ2 2 Ϫ O 2 2 Ϫ2 Ϫ2 T S
LK
4. Quadrilateral ABCD has vertices A(4, 3), B(4, −3), C(−4, −3), and D(−4, 3). a. Find a coordinate rule that describes a stretch that, when applied to ABCD, results in an image that is a square. Explain your reasoning. b. Find a coordinate rule that describes a compression that, when applied to ABCD, results in an image that is a square. Explain your reasoning. 5. Create Representations to Communicate Mathematical Ideas (1)(E) A classmate dilates a figure in the coordinate plane by a scale factor greater than 1 and then compresses the resulting figure vertically by a factor between 1 and 0. a. Write a coordinate rule that describes the dilation. b. Write a coordinate rule that describes the vertical compression. Find the vertices of the image of each figure after the given composition of transformations. Then graph the image.
6. a dilation with scale factor 7. a dilation with scale factor 1 1 2 2 centered at the origin, centered at point N, followed followed by the transformation by a vertical stretch with 1 (x, y) S (2x, y) factor 3 y NL y M Ϫ4 Ϫ2 O x 2 L N O 2 4 x M
368 Lesson 8-8 Other Non-Rigid Transformations △A″B″C″ is a transformation of △ABC. Determine the coordinates for △ABC before each composition of transformations. Then graph △ABC. 8. a dilation with scale factor 9. a dilation with scale factor 3 3 centered at the origin, centered at point C, followed by followed by the transformation a horizontal compression with 2 1 (x, y) S (3x, y) factor 2 A y y C x O B 2 4 Ϫ2 C x Ϫ2 O B A
10. Analyze Mathematical Relationships (1)(F) A rectangle in the coordinate plane is dilated by a scale factor of 3 and then stretched horizontally by a factor of 2. Explain how to find the coordinates of the vertices of the preimage if you know the coordinates of the vertices of the image. Describe a sequence of transformations that maps quadrilateral EFGH to quadrilateral E′F′G′H′. 11. y 12. y F F F 6 6 E E 4 G E 4 G G
2 F 2 H EG x x Ϫ4 Ϫ2 O H = H 4 -4 -2 O H 4
13. Apply Mathematics (1)(A) A rancher’s plan to 14 expand some stables is shown on the coordinate grid. The rancher plans to make the stables larger 12 and move them across the ranch. Describe a sequence of transformations that will move the 10 stables to the outlined location on the rancher’s plan. 8 14. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) △PQR has vertices 6 P(-2, -2), Q(2, -1), and R(-4, -3), and △P‴Q‴R‴ 4 has vertices P‴(-3, -8), Q‴(5, -4), and R‴(-7, -12). A dilation with scale factor 4, followed by a second 2 transformation, followed by a horizontal compression with factor 1 maps △PQR to △P‴Q‴R‴. Describe 2 0 2 4 6 8 10 12 14 the second transformation using both words and a coordinate rule.
PearsonTEXAS.com 369 15. A computer game programmer is testing a transformations game. Describe a possible sequence of transformations the programmer could use to map Figure A onto Figure B. Are the figures congruent? Explain.
Game: Galactic Grid 6 B Difficulty: 12:30 PM Medium 4 Tester: 2 A GamerGal8 0 2 4 6
16. Explain Mathematical Ideas (1)(G) A sequence of a rigid transformation followed by a non-rigid transformation is applied to a non-rectangular figure in the coordinate plane, with the result that the angle measures in the preimage and the final image are equal. What does this tell you about the rigid transformations in the sequence? What does it tell you about the non-rigid transformations? Explain your reasoning.
TEXAS Test Practice
17. Which transformation maps △ABC to △A′B′C′? A y 2 2 A. (x, y) S x, y (3 3 ) A 1 1 B. (x, y) S (3 x, 3 y) 2 1 2 C. (x, y) S x, y (3 3 ) B x 1 2 C D. (x, y) S (3 x, 3 y) O B C 2 4 18. Which sequence of transformations does not preserve congruence? F. a dilation followed by a rotation G. a reflection followed by a rotation H. a translation followed by a translation J. a translation followed by a reflection 19. If ∠1 and ∠2 are vertical angles, which of the following statements must be true? A. m∠1 6 m∠2 C. m∠1 + m∠2 = 90 B. m∠1 = m∠2 D. m∠1 + m∠2 = 180 20. Explain how to write a coordinate proof to show that two lines in the coordinate plane are perpendicular.
370 Lesson 8-8 Other Non-Rigid Transformations Topic 8 Review
TOPIC VOCABULARY
• angle of rotation, p. 332 • dilation, p. 356 • orientation, p. 326 • rotation, p. 332 • center of dilation, p. 356 • enlargement, p. 356 • point symmetry, p. 338 • rotational symmetry, p. 338 • center of rotation, p. 332 • glide reflection, p. 344 • preimage, p. 318 • scale factor of a dilation, • composition of • image, p. 318 • ratio, p. 356 p. 356 transformations, p. 318 • line of reflection, p. 326 • reduction, p. 356 • stretch, p. 364 • compression, p. 364 • line of symmetry, p. 338 • reflection, p. 326 • transformation, p. 318 • congruence • line symmetry, p. 338 • reflectional symmetry, • translation, p. 319 transformation, p. 350 • non-rigid transformation, p. 338 • congruent, p. 350 p. 356 • rigid transformation, p. 318
Check Your Understanding Choose the correct term to complete each sentence. 1. A(n) ? is a change in the position, shape, or size of a figure. 2. A(n) ? is a composition of a translation and a reflection. 3. In a(n) ? , all points of a figure move the same distance in the same direction. 4. A(n) ? is a transformation that preserves distance and angle measure.
8-1 Translations
Quick Review Exercises F A transformation of a geometric figure is a change in its 5. a. A transformation maps L O position, shape, or size. ZOWE onto LFMA. Does the Z A translation is a rigid transformation that maps all points transformation appear to be a rigid transformation? Explain. of a figure the same distance in the same direction. E W In a composition of transformations, each b. What is the image of ZE? A M transformation is performed on the image of the What is the preimage of M? preceding transformation. 6. △RST has vertices R(0, -4), S( -2, -1), and T( -6, 1). Graph T 4, 7 (△RST). 6- 7 hsm11gmse_09cr_t10516.ai Example 7. Write a rule to describe a translation 5 units left and 10 units up. What are the coordinates of T6-2, 37(5, -9)? 8. Find a single translation that has the same effect Add -2 to the x-coordinate, and 3 to the y-coordinate. as the following composition of translations. A(5, -9) S (5 - 2, -9 + 3), or A′(3, -6). T6-4, 77 followed by T63, 07
PearsonTEXAS.com 371 8-2 Reflections
Quick Review Exercises The diagram shows a reflection across line r. A reflection Given points A(6, 4), B( 2, 1), and C(5, 0), graph △ABC is a rigid transformation that preserves distance and angle and each reflection image. measure. The image and preimage of a reflection have 9. R (△ABC) opposite orientations. x-axis 10. Rx = 4(△ABC)
11. Ry = x(△ABC) 12. Copy the diagram. Then draw R (BGHT). Label r y-axis the vertices of the image, using prime notation.
y Example 4 B Use points P(1, 0), Qhsm11gmse_09cr_t10508.ai(3, 2), and R(4, 0). What is R (△PQR)? y-axis T G y x Graph PQR. Find P , Q , △ ′ ′ RЈ PЈ P R x Ϫ4 Ϫ2 4 and R′ such that the O 5 y-axis is the perpendicular H Ϫ2 bisector of PP′, QQ′, and QЈ Q Ϫ4 RR′. Draw △P′Q′R′.
8-3 Rotations hsm11gmse_09cr_t10511.ai
Quick Review Exercises geom12_se_ccs_c09cr_t03.ai The diagram shows a rotation of x° about point R. A 13. Copy the diagram below. Then draw r(90°, P)(△ZXY). rotation is a rigid transformation in which a figure and its Label the vertices of the image, using prime notation. image have the same orientation. Z X P
Њ R x Y
14. What are the coordinates of r(180°, O)(-4, 1)? 15. WXYZ is a quadrilateral with vertices W(3, 1), hsm11gmse_09cr_t10512.ai- Example X(5, 2), Y(0, 8), and Z(2, -1). Graph WXYZ GHIJ has vertices G(0, 23), H(4, 1), I(21, 2), and and r(270°, O)(WXYZ). hsm11gmse_09cr_t10509.ai J(25, 22). What are the vertices of r(90, O)(GHIJ)?
Use the rule r(90°, O)(x, y) = (-y, x).
r(90°, O)(G) = (3, 0) r(90°, O)(H) = (-1, 4) r(90°, O)(I) = (-2, -1) r(90°, O)(J) = (2, -5)
372 Topic 8 Review 8-4 Symmetry
Quick Review Exercises A figure has reflectional symmetry or line symmetry if Tell what type(s) of symmetry each figure appears to there is a reflection for which it is its own image. have. If it has reflectional symmetry, sketch the figure A figure that has rotational symmetry is its own image for and the line(s) of symmetry. If it has rotational some rotation of 180° or less. symmetry, state the angle of rotation. A figure that has point symmetry has 180° rotational 16. 17. 18. symmetry.
Example 19. How many lines of symmetry does an isosceles How many lines of symmetry does an equilateral trapezoid have? triangle have? 20. What type(s) of symmetry does a square have? An equilateral triangle reflects onto itself across each of its three medians. The triangle has three lines of symmetry.
8-5 Compositions of Rigid Transformations
Quick Review Exercises ഞ m A rigid transformation preserves distance and angle 21. Sketch and describe the result of measure. You have learned about translations, reflections, reflecting E first across line / and E and rotations, which are all rigid transformations. A then across line m. composition of rigid transformations is also a rigid transformation. All rigid transformations can be expressed Each figure is the image of the figure below. Tell whether as a composition of reflections. their orientations are the same or opposite. Then classify the transformation. hsm11gmse_09cr_t10524.ai The diagram shows a glide reflection N N of N. A glide reflection is a rigid gl n e transformation in which a figure and a its image have opposite orientations. N 22. 23. gl 24. e
gle n e l
a
n g
a n Example a hsm11gmse_09cr_t10525.ai Describe the result of reflecting P first hsm11gmse_09cr_t10522.ai C m across line O and then across line m. 50Њ 25. △TAM has vertices T (0, 5), A(4, 1), and M(3, 6). hsm11gmse_09cr_t10527.ai ഞ Find the image of Ry = -2 ∘ T(-4, 0)(△TAM). A composition of two reflections across P hsm11gmse_09cr_t10526.ai intersecting lines is a rotation. The angle hsm11gmse_09cr_t10528.ai of rotation is twice the measure of the acute angle formed by the intersecting lines. P is rotated 100° about C. hsm11gmse_09cr_t10523.ai
PearsonTEXAS.com 373 8-6 Congruence Transformations
Exercises Quick Review L y Two figures are congruent if and only if there is a sequence 26. In the diagram at the right, of rigid transformations that maps one figure onto the other. △LMN ≅ △XYZ. Identify a congruence transformation M N x that maps △LMN onto △XYZ. Z Example y W 27. Graphic designers use some X R (TGMB) KWAV. What G y-axis fonts because they have pleasing Y are all of the congruent angles proportions or are easy to read from and all of the congruent sides? T M A K far away. The letters p and d above B V A reflection is a congruence x are used on a sign that has a special font. Are the letters transformation, so TGMB ≅ KWAV, congruent? If so, describe a congruence transformation and corresponding angles and that maps one onto the other. If not, explain why not. corresponding sides are congruent. ∠T ≅ ∠K, ∠G ≅ ∠W , ∠M ≅ ∠A, and ∠B ≅ ∠V TG = KW , GM = WA, MB = AV, and TB = KV p d geom12_se_ccs_c09cr_t06.ai
8-7 Dilations geom12_se_ccs_c09cr_t05.ai geom12_se_ccs_c09cr_t07.ai Quick Review Exercises The diagram shows a dilation with center C and scale factor 28. The blue figure is a dilation image of the black figure. n. Dilations preserve angle measures. The center of dilation is O. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor.
C y a na 4 In the coordinate plane, if the origin is the center of a dilation with scale factor n, then P(x, y) P (nx, ny). S ′ x hsm11gmse_09cr_t10510.ai Ϫ2 O 2 4 Example Graph the polygon with the given vertices. Then graph The blue figure is a dilation image of its image for a dilation with center (0, 0) and the given the black figure. The center of dilation scale factor. 4 is A. Is the dilation an enlargement or a hsm11gmse_09cr_t10521.ai reduction? What is the scale factor? 29. M( -3, 4), A( -6, -1), T (0, 0), H(3, 2); scale factor 5 1 30. F( -4, 0), U(5, 0), N( -2, -5); scale factor The image is smaller than the preimage, 2 2 so the dilation is a reduction. The scale A 31. A dilation maps △LMN onto △L′M′N′. LM = 36 ft, image length 2 2 1 LN = 26 ft, MN = 45 ft, and L′M′ = 9 ft. Find L′N′ factor is original length = 2 + 4 = 6, or 3. and M′N′. hsm11gmse_09cr_t10517.ai
374 Topic 8 Review 8-8 Other Non-Rigid Transformations
Quick Review Exercises A horizontal stretch/compression is any transformation Determine and graph P′Q′R′S′, the image of PQRS after (x, y) S (ax, y) for a 7 0. each transformation or composition of transformations. A vertical stretch/compression is any transformation P y Q (x, y) S (x, by) for b 7 0. A non-rigid transformation that stretches or compresses a x O figure by different amounts in different directions does not S R preserve congruence. 32. (x, y) S (2x, y) 1 Example 33. (x, y) S (x, 2y) △ABC has vertices A(-2, -1), B(1, 1), and C(2, -1). 34. a dilation centered at the origin with scale factor 2 What are the coordinates after the transformation 1 followed by the transformation (x, y) S x, y (x, y) S (x, 2y)? (4 ) 35. a dilation with scale factor 1 centered at point Q y 2 followed by the translation (x, y) S (x - 2, y + 1) 2 B x 36. If PQRS is the image of EFGH after a dilation of scale -2O 2 1 A C factor 3 followed by the transformation (x, y) S (2x, y), what are the coordinates of EFGH? A(-2, -1) S A′(-2, -2) 37. Describe a sequence of transformations that will map B(1, 1) S B′(1, 2) PQRS to JKLM with vertices J(-6, 1), K(6, 1), L(6, -2), C(2, -1) S C′(2, −2) and M(-6, -2).
PearsonTEXAS.com 375 Topic 8 TEKS Cumulative Practice
Multiple Choice 6. What type of symmetry does the figure have? Read each question. Then write the letter of the correct answer on your paper. 1. In a right triangle, which point lies on the hypotenuse? A. incenter C. centroid B. orthocenter D. circumcenter F. 60° rotational symmetry 2. In △LMN, P is the centroid and LE = 24. What is PE? G. 90° rotational symmetry M H. line symmetryhsm16_gmhh_08cp_t012.ai D E J. point symmetry P 7. Which conditions allow you to conclude that a L N F quadrilateral is a parallelogram? F. 8 H. 10 A. one pair of sides congruent, the other pair of sides parallel G. 9 J. 16 B. perpendicular, congruent diagonals 3. Whathsm11gmse_09cu_t08646.ai is the sum of the angle measures of a 32-gon? C. diagonals that bisect each other A. 3200° C. 5400° D. one diagonal bisects opposite angles B. 3800° D. 5580° 8. Write the horizontal stretch rule that maps 4. The diagonals of rectangle PQRS intersect at H. What is P( 1, 2) to P ( 3, 2). the length of QS? - ′ - F. (x, y) ( 3x, y) P Q S - 3x ϩ 5 G. (x, y) S (x, 3y) 4x H Ϫ 1 H. (x, y) S (3x, y)
S R J. (x, y) S (-3x, -y) F. 6 H. 23 9. What type of symmetry does the figure have? G. 12 J. 46
5. hsm11gmse_09cu_t08647.aiThe vertices of ▱ABCD are A(1, 7), B(0, 0), C(7, -1), and D(8, 6). What is the perimeter of ▱ABCD?
A. 50 C. 200 A. reflectional symmetry 2 B. 100 D. 20 2 B. rotational symmetry 2 C. point symmetryhsm11gmse_09ct_t10571.ai D. no symmetry
376 Topic 8 TEKS Cumulative Practice 10. If you are given a line and a point not on the line, what Constructed Response is the first step to construct the line parallel to the given line through the point? 15. What is the value of x for which p } q? F. Construct an angle from a point on the line to the p given point. 115Њ G. Draw a straight line through the given point. q ͑2x ϩ 5͒Њ H. Draw a ray from the given point that does not intersect the line. 16. △DEB has vertices D(3, 7), E(1, 4), and B(-1, 5). In J. Label a point on the given line, and draw a line which quadrant(s) is the image of r(270°, O)(△DEB)? through that point and the given point. Draw a diagram. hsm11gmse_09cu_t08648.ai 11. Which quadrilateral must have congruent diagonals? 17. In △ABC below, AB ≅ CB and BD # AC. Prove that A. kite C. parallelogram △ABD ≅ △CBD. B. rectangle D. rhombus B
Gridded Response 12. What is the measure of ∠H?
A F 50Њ G A D C 35Њ 18. Is △ABC a right triangle? Justify your answer. B C H y 13. What is the area of the square, in square units? 4 A hsm11gmse_09cu_t08651.ai y hsm11gmse_09cu_t08649.ai x Ϫ4 O 4 2 x B C Ϫ4 Ϫ2 O 4 19. LMNO has vertices L( -4, 0), M( -2, 3), N(1, 1), and Ϫ3 O( -1, -2). RSTV has vertices R(1, 1), S(3, -2), T(6, 0), and V(4, 3). Graph the two quadrilaterals. 14. In ▱PQRS, what is the value of x? hsm11gmse_09cu_t08652.aiIs LMNO ≅ RSTV? If so, write the rule for the congruence transformation that maps LMNO to Q R RSTV. If not, explain why not. hsm11gmse_09cu_t08729.aixЊ 84Њ O
P 22Њ S
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