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 DIGITAL APPS PRINT and eBook Access Your Homework . ONLINE HOMEWORK You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. HOMEWORK TUTOR APP YOUR DIGITAL Do your homework anywhere! You can access the Practice and RESOURCES Application Exercises, as well as Virtual Nerd tutorials, with this PearsonTEXAS.com Homework Tutor app, available on any mobile device. STUDENT TEXT AND HOMEWORK HELPER Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 316 Topic 8 Transformational Geometry The Perplexing Polygon Look around and you will probably see shapes and patterns everywhere you look. Math The tiles on a floor are often all the same shape and fit together to form a pattern. The petals Act Math Act Act - - on a flower frequently create 3 3 a repeating pattern around the center of the flower. When you look at snowflakes under a microscope, you’ll notice that they are made up of repeating three-dimensional crystals. Think about this as you watch this 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . VOCABULARY ONLINE LEARNING ANIMATIONS INTERACTIVE MATH TOOLS You’ll find definitions of math You can also access all of the These interactive math tools terms in both English and stepped-out learning animations give you opportunities to Spanish. All of the terms have that you studied in class. explore in greater depth audio support. key concepts to help build understanding. INTERACTIVE EXPLORATION STUDENT COMPANION VIRTUAL NERD You’ll have access to a robust Refer to your notes and Not sure how to do some of assortment of interactive solutions in your Student the practice exercises? Check explorations, including Companion. Remember that out the Virtual Nerd videos interactive concept explorations, your Student Companion is also for stepped-out, multi-level dynamic activitites, and topic- available as an ACTIVebook instructional support. level exploration activities. accessible on any digital device. PearsonTEXAS.com 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. hsm11gmse_0901_t07520.ai 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) hsm11gmse_0901_t07524.ai 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 EFGH. 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 hsm11gmse_0901_t07526.ai 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 PQRS? 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.