Mathematics
Congruence Transformations
Congruence transformations in mathematics refer to transformations that preserve the shape and size of a geometric figure. These transformations include translations, rotations, reflections, and dilations. When a figure undergoes a congruence transformation, its corresponding parts maintain the same relative positions and lengths, resulting in a congruent figure.
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11 Key excerpts on "Congruence Transformations"
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
567 10 SECTION 10.1 Congruence Transformations SECTION 10.2 Symmetry and Tessellations SECTION 10.3 Similarity Geometry as Transforming Shapes In Chapter 9 we focused on understanding measurement of shapes, and in this chapter, we are going to look at what happens when we transform shapes. This will help us better understand the patterns that emerge when shapes are put together, as in quilts, floor patterns, art, and other situations. Look at the pictures in Figures 10.1 to 10.3. What do you see? What did the artist who created these have to think about? Note your ideas before reading on. Figure 10.1 Azat1976/Shutterstock.com Figure 10.2 Margrit Hirsch/Shutterstock.com Figure 10.3 OKing/Shutterstock.com 568 CHAPTER 10 Geometry as Transforming Shapes n n Transformations Before we get into the mathematical aspect of transformations, what do you think when you hear this word? The interesting examples offered by my students include the transformation of a frog into a prince in fairy tales and the transformation of a caterpillar to a butterfly. You are in the process of transforming from a student to a teacher. Virtually all examples of trans- formation give a sense of movement and change. In mathematics, there are many kinds of transformations. Explorations Manual 10.1, 10.2, 10.3 What do you think? l Where have you heard the words translation, reflection, and rotation in everyday life? l How might you describe the translation, reflection, or rotation of a three-dimensional figure in space? l How are the operations translation, reflection, and rotation like the operations addition, subtraction, multiplication, and division? Congruence Transformations SECTION 10.1 INVESTIGATION 10.1a How Do Things Move? A. Take an object such as your cell phone, a tangram triangle, an index card, and so on, and explore all the different ways you can move that object. Describe each unique type of movement and give it a name that makes sense to you.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Figure 10.21 Figure 10.22 Figure 10.23 We have examined four fundamental ways in which we can transform a geometric figure: translation, reflection, rotation, and glide reflection. We have uncovered connec-tions among these transformations. For example, performing two specific reflections is equivalent to performing a specific rotation. We have discovered that there are many similarities between numbers and shapes: They have important subsets, we do opera-tions on them, we can make tables for the operations, we can decompose numbers and shapes, and there are important properties that help us better understand our operations on numbers and shapes. These investigations of geometric transformations of two-dimensional figures in a plane can be extended. What if we examined transformations of three-dimensional objects? These investigations occupy the attention of many mathematicians and have applications in other fields. For example, the way in which atoms are packed helps to determine the properties of a compound. SUMMARY 10.1 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 574 CHAPTER 10 Geometry as Transforming Shapes Unless otherwise noted, all content on this page is © Cengage Learning 10.1 Exercises M A T H M A T H X X 4. The figure below shows quadrilateral ABCD and a line of reflection. Determine the coordinates of A r , B r , C r , and D r using only reasoning (that is, without folding). Explain your reasoning. 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 x y r A D C B 5.
- Sarah Schuhl, Timothy D. Kanold, Bill Barnes, Darshan M. Jain, Matthew R. Larson, Brittany Mozingo(Authors)
- 2020(Publication Date)
- Solution Tree Press(Publisher)
The next chapter shows examples of a high school geometry unit for transformations and congruence. Read through the examples to see possible standards teachers expect students to learn next year.
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CHAPTER 4
Geometry Unit: Transformations and Congruence
G eometry students learn to apply theorems, postulates, and definitions to proofs, mathematical problems, and real-world problems as they grow their mathematical reasoning skills. Through your intentional unit design, your geometry team has an opportunity to build student confidence, self-efficacy, and a productive disposition with transformations (a type of function) and congruence.In eighth grade, students develop an understanding of congruence by experimenting with the properties of rotations, reflections, and translations, also known as rigid motions. Students use physical models, transparencies, coordinate grids, and geometry software to understand that a two-dimensional figure is congruent to another if its preimage can be mapped onto an image using a sequence of rigid motions.In algebra 1, students explore transformations by analyzing the size, shape, and position effects on the graph of the parent functions for the linear and quadratic functions, f(x) = x and f(x) = x2 , when f(x) is replaced by af(x), f(x) + d, f(x — c), and f(bx) for specific values a, b, c, and d. Next year, in algebra 2, students will analyze effects of transformations on the graphs of additional parent functions such as exponential, rational, radical, logarithmic, and trigonometric functions.Understanding what students have learned in eighth grade and algebra 1, as well as what they will learn in algebra 2, helps frame the learning needed in geometry as part of the greater story arc for student learning in high school mathematics.In geometry, students use their experiences with geometric transformations to develop formal proofs for congruence. Students represent and perform transformations in a plane using various tools, describe transformations as functions with ordered pairs for domain and range, and compare transformations that preserve distance and angle to those that do not. Students then use the definition of congruence in terms of rigid motion to decide and explain if two shapes are congruent. Additionally, students explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence preserved by rigid motions.
eBook - PDF- H. S. M. Coxeter, S. L. Greitzer(Authors)
- 1967(Publication Date)
- American Mathematical Society(Publisher)
C H A P T E R 4 Transformations By faith E n d was translated that he should not see death; and was not found, because God had translated him: for before his translation he had this testimony, that he pleased God. Hcbmus, 11 :5 In a remark at the end of Section 1.6, we obtained the right angle between F D and OB (Figure 1.6A) by rotating the perpendicular lines H D and CB through equal angles a about D and B, respectively. In the preamble to Theorem 1.71, we observed that the two similar triangles ABC and A'B'C' have the same centroid and that, since their orthocenters are H and 0, A H = 20A'. Finally, in the remark after Theorem 1.81, we used a half-turn to interchange the orthocenters of the two congruent triangles A'B'C' and KLM. The rotation, dilatation, and half-turn are three instances of a transjmation which (for our present purposes) means a mapping of the whole plane onto itself so that every point P has a unique image PI, and every point Q' has a unique prototype Q. This idea of a mapping figures prominently in most branches of mathematics; for instance, when we write y = f(x) we are mapping the set of values of x on the set of corresponding values Euclidean geometry is only one of many geometries, each having its own primitive concepts, axioms, and theorems. Felix Klein, in his inaugural address at Erlangen in 1872, proposed the classification of geometries according to the groups of transformations that can be applied without changing these concepts, axioms, and theorems. In particular, Euclidean geometry is characterized by the group of simi- larities; these are angle-preserving transformations. An important special case of a similarity is an isometry. This is a Zength-preserving transforma- tion such as a rotation or, in particular, a half-turn. Isometries are at the bottom of the familiar idea of congruence: two figures are congruent if and only if one can be transformed into the other by an isometry. 80 of y.
eBook - ePubMaking Sense of Mathematics for Teaching Grades 6-8
(Unifying Topics for an Understanding of Functions, Statistics, and Probability)
- Edward C. Nolan, Juli K. Dixon(Authors)
- 2016(Publication Date)
- Solution Tree Press(Publisher)
The pre-image and image, however, are congruent shapes. The pre-image has turned about the center of rotation to determine the image. How can you determine where the points of the image will appear? The measure of ∠ CHC ' is 45°, with a clockwise rotation. This tells you that points A ' and B ' will fall on the rays that would create 45° angles as ∠ AHA ' and ∠ BHB '. Students will sometimes encounter a challenge when the center of rotation is separate from the shape, as in this case. It is important for students to consider how the shape rotates about the center and maintains the size and shape of the figure as well as the distance between the shape and the center of rotation. Figure 5.29: Rotation of Δ ABC. These three transformations connect to congruency in that they are rigid motion transformations. They maintain the original pre-image, so the image is identical to the pre-image in size and shape. In all three cases illustrated in figures 5.27, 5.28, and 5.29, the result of the transformation preserves the original size and shape of the figure, although not necessarily the orientation. As students engage with geometric transformations, they use rigid motion transformations to show that shapes are congruent to each other. Students must be able to justify which transformation or combination of transformations would map the original figure onto the congruent shape. While these first three types of transformations preserve the original shape, the transformation of dilation does not create a congruent shape. Instead, it creates a shape that is similar to the original shape—the same shape but drawn to a different scale (see figure 5.30)
eBook - PDF- Ronald N. Umble, Zhigang Han(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 4 Translations, Rotations, and Reflections In grades 9–12 all students should apply transformations and use symmetry to...understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function nota-tion, and matrices . Principles and Standards for School Mathematics National Council of Teachers of Mathematics (2000) Studying geometric transformations provides opportunities for learners to describe patterns, discover basic features of isometries, make generalizations, and develop spatial competencies . H. Bahadir Yanick, Math. Educator Anadolu University In this chapter we consider three important families of isometries and investigate some of their properties. We motivate each section with an ex-ploratory activity. The instructions for these activities are generic and can be performed using any software package that supports geometric constructions. The Geometer’s Sketchpad commands required by these activities appear in the appendix at the end of the chapter. 4.1 Translations A translation of the plane is a transformation that slides the plane a finite distance in some direction. Exploratory activity 1, which follows below, uses the vector notation in the following definition: 45 46 Transformational Plane Geometry Definition 97 A vector v = a b is a quantity with norm (or magnitude) k v k := √ a 2 + b 2 and direction Θ defined by the equations k v k cos Θ = a and k v k sin Θ = b . The values a and b are called the x -component and y -component of v , respectively. The vector 0 = 0 0 , called the zero vector , has magnitude 0 and arbitrary direction. Given vectors v = a b and w = c d , the dot product is defined to be v · w := ac + bd . Thus v · v = a 2 + b 2 = k v k 2 . If P = x y and Q = x 0 y 0 are points, the vector PQ = x 0 -x y 0 -y is called the position vector from P to Q, and P and Q are called the initial and terminal points of PQ .
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
iStockphoto.com/asiseeit; ostill/Shutterstock.com The dance instructor is teaching the man the basic steps in the dance called the salsa. Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 14.1 Translations and Rotations 531 14.1 Translations and Rotations Understand and identify the four basic types of transformations. Understand and use the concept of a translation. Understand and use the concept of a rotation. Basic Types of Transformations A transformation changes a figure into another figure. The new figure is called the image. There are four basic types of transformations. 1. Translation (Section 14.1) 2. Rotation (Section 14.1) Slide Turn 3. Reflection (Section 14.2) 4. Dilation (Section 14.3) Flip Dilate The first three types of transformations above (translations, rotations, reflections) are rigid transformations because they preserve the size and shape of the original figure. In other words, the image is congruent to the original figure. Standards Grades 6–8 Geometry Students should understand congruence and similarity using physical models, transparencies, or geometry software. EXAMPLE 1 Identifying Transformations in Art What types of transformations are shown in the tessellation? Explain your reasoning. SOLUTION There are translations and rotations. Rotation Translation There are no reflections or dilations in this tessellation. For the tessellation shown in Example 1, you can convince your students that there are no dilations because all of the lizards are congruent.
- Ronald Goldman(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 4 Af fi ne Transformations for Satan himself is transformed into an angel of light. – 2 Corinthians 11:14 4.1 Transformations Transformations are the lifeblood of geometry. Euclidean geometry is based on rigid motions — translation and rotation — transformations that preserve distances and angles. Congruent triangles are triangles where corresponding lengths and angles match. Transformations generate geometry. The turtle uses translation (FORWARD), rotation (TURN), and uniform scaling (RESIZE) to generate curves by moving about on the plane. We can also apply translation (SHIFT), rotation (SPIN), and uniform scaling (SCALE) to build new shapes from previously de fi ned turtle programs. These three transformations — translation, rotation, and uniform scaling — are called conformal transformations . Conformal transformations preserve angles, but not distances. Similar triangles are triangles where corresponding angles agree, but the lengths of corresponding sides are scaled. The ability to scale is what allows the turtle to generate self-similar fractals like the Sierpinski gasket. In Computer Graphics, transformations are employed to position, orient, and scale objects as well as to model shape. Much of elementary Computational Geometry and Computer Graphics is based upon an understanding of the effects of different fundamental transformations. The transformations that appear most often in two-dimensional Computer Graphics are the af fi ne transformations. Af fi ne transformations are composites of four basic types of transformations: translation, rotation, scaling (uniform and nonuniform), and shear. Af fi ne transformations do not necessarily preserve either distances or angles, but af fi ne transformations map straight lines to straight lines and af fi ne transformations preserve ratios of distances along straight lines (see Figure 4.1). For example, af fi ne transformations map midpoints to midpoints.
eBook - PDF- David A. Brannan, Matthew F. Esplen, Jeremy J. Gray(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
You will meet some examples of this in Section 2.5 . 2.3.2 The Fundamental Theorem of Affine Geometry The algebraic approach can also be used to investigate whether there is an affine transformation which maps one given figure onto another. Recall that if there is such a transformation, then the two figures are said to be affine-congruent . This concept of congruence is important because, as we explained in Section 2.1 , figures that are affine-congruent to each other share the same affine properties. In this subsection we prove the remarkable result that all triangles are affine-This is very different to Euclidean geometry, where two triangles are congruent only if they have the same shape and size. congruent and therefore share the same affine properties. In fact, since a triangle is completely determined by its three vertices, the congruence of tri-angles follows from the so-called Fundamental Theorem of Affine Geometry which states that any three non-collinear points can be mapped to any other three non-collinear points by an affine transformation. First, recall that in Subsection 2.2.3 we described how the points (0, 0), (1, 0) and (0, 1) in R 2 can be mapped to any three non-collinear points P , Q There the mapping was constructed in a geometric manner. In this subsection we construct the mapping algebraically. and R by an affine transformation. This transformation is unique in the sense that it is completely determined by the choice of P , Q and R . The following example should remind you of how such transformations are constructed. Example 2 Determine the affine transformation which maps the points (0, 0), (1, 0) and (0, 1) to the points (3, 2), (5, 8) and (7, 3), respectively. Solution Let t be the affine transformation given by t : x y → a b c d x y + e f . (6) Since t ( 0, 0 ) = ( 3, 2 ) , it follows from (6) that e = 3 and f = 2. Next, t ( 1, 0 ) = ( 5, 8 ) , so it follows from (6) that 5 8 = a b c d 1 0 + 3 2 = a c + 3 2 .
eBook - ePub- Hermann Weyl(Author)
- 2013(Publication Date)
- Dover Publications(Publisher)
so that B falls on A, and A falls on B). On the other hand, a congruent transformation which transforms A into itself, and all points to the right of A into points to the right of A, and all points to the left of A into points to the left of A, leaves every point of the straight line undisturbed. The homogeneity of the straight line is expressed in the fact that the straight line can be placed upon itself in such a way that any point A of it can be transformed into any other point A ′ of it, and that the half to the right of A can be transformed into the half to the right of A ′, and likewise for the portions to the left of A and A ′ respectively (this implies a mere translation of the straight line). If we now introduce the equation AB′ B ′ for the points of the straight line by interpreting it as meaning that AB is transformed into the straight line A ′‘ B ′ by a translation, then the same things hold for this conception as for time. These same circumstances enable us to introduce numbers, and to establish a reversible and single correspondence between the points of a straight line and real numbers by using a unit of length OE. Let us now consider the group of congruent transformations which leaves the straight line g fixed, i.e. transforms every point of g into a point of g again. We have called particular attention to rotations among these as having the property of leaving not only g as a whole, but also every single point of g unmoved in position. How can translations in this group be distinguished from twists ? I shall here outline a preliminary argument in which not only the straight line, but also the plane is based on a property of rotation. Two rays which start from a point O form an angle. Every angle can, when inverted, be superposed exactly upon itself, so that one arm falls on the other, and vice versa. Every right angle is congruent with its complementary angle
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
e. An isometry preserves distance and angle measure. f. An isometry preserves orientation. g. A size transformation preserves angle measure and ratios of length. h. A similarity transformation is a size transformation followed by an isometry. 2. List the properties that are preserved by isometries but not size transformations. 3. Which of the transformations leaves exactly one point fixed when performed on the entire plane? 4. Which of the following transformations map the segment AB to ′ ′ A B in such a way that AB A B || ? ′ ′ (There may be more than one correct answer.) a. Translation b. Rotation c. Reflection d. Glide reflection e. Size transformation 5. Which of the following transformations map the segment AB to ′ ′ A B in such a way that AA BB ′ ′ || ? (There may be more than one correct answer.) a. Translation b. Rotation c. Reflection d. Glide reflection e. Size transformation 6. Each of the following notations describes a specific trans- formation. Identify the general type of transformation cor- responding to each notational description. a. S Q M, ( ) 3 b. R Q N b , ( ) c. M Q n ( ) d. M T Q n xy ( ( )) e. T Q xy ( ) Skill 7. For each of the following, trace the figure onto a piece of paper and perform the indicated transformation on the quadrilateral. a. Reflect about line m. b. Rotate 90° about point O . c. Translate parallel to the directed line segment MN. CHAPTER TEST 874 Chapter 16 Geometry Using Transformations 8. Find the following points on the square lattice. a. T P AB ( ) b. R P C, ( ) 90° c. M P OC ( ) d. T M P PB OC ( ( )) 9. Trace the following grid and triangle on a piece of paper and find S ABC O, ( ) 3 n . 10. Determine which of the following types of symmetry apply to the tessellation shown here: translation, rotation, reflection, glide reflection (assume that the tessellation fills the plane). 11. Describe the following isometries as they relate to the tri- angles shown.
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