Mathematics
Reflection in Geometry
Reflection in geometry is a transformation that flips a figure over a line, known as the line of reflection. The original figure and its reflection are mirror images of each other, with the same size and shape. This concept is fundamental in understanding symmetry and congruence in geometric shapes and patterns.
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10 Key excerpts on "Reflection in Geometry"
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
The most succinct way to describe this is by defining the line of reflection, which is the line that the figure is being flipped across. One way to look at reflections is through paper folding. Trace the DDOG and the line below from Figure 10.12 onto a blank sheet of paper, and then fold the paper at the line m. 574 CHAPTER 10 Geometry as Transforming Shapes this several times. Play the “what-if” and “is it possible” games: What if the line of reflection goes through the figure? Is it possible to have the line of reflection go right through the figure? B. Properties of reflections. What is true for all reflections is that if we connect any point on the original figure with the corresponding point on the reflected figure, the line of reflection is the perpendicular bisector of that line segment (Figure 10.14). That is, 5 A X XA, and AA and line l are perpendicular. This realization leads to a more formal definition of reflection. A reflection is a transformation that maps a figure so that a line, called the line of reflection, is the perpendicular bisector of every line segment joining a point on the figure and the cor- responding point on the reflected figure. The two-dimensional figures we just looked at were reflected across a line. What if we were reflecting a three-dimensional object, like yourself? Then we would have a plane of reflection. When you reflect yourself across a mirror, the mirror is the plane of reflection. A A' X l Figure 10.14 n n Rotations Consider the word “rotation.” What does it mean? In what other contexts do we use the word? The rotation of a car wheel, hands on a clock, or of Earth on its axis are all familiar to us. INVESTIGATION 10.1e Communicating Rotations For this you will need some sticky notes, graph paper, and a partner. This is similar to Inves- tigation 10.1b where you both started with two sticky notes on top of each other on a piece of graph paper. Put up a book or notebook in between you and your partner.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
mathematicians and half were art professors. My job was to keep them from killing each other. Seriously, part of my job was to help them recognize what they had in common. One of the many things they had in common was a love of patterns and a love of break-ing patterns. SuperStock 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. 594 CHAPTER 10 Geometry as Transforming Shapes Unless otherwise noted, all content on this page is © Cengage Learning If there is a plane that can be drawn through a three-dimensional figure so that one half of the figure is a mirror image of the other, then the figure has reflection symmetry. The plane is called the plane of symmetry . Similarly, when we defined rotation symmetry with two-dimensional figures, we used a point (a one-dimensional object). We define rotation symmetry with three- dimensional figures using a line (a two-dimensional object). If there is a line that can be drawn, around which a three-dimensional figure can be rotated so that it coincides with itself, then the figure has rotation symmetry. The line about which the figure is rotated is called the axis of symmetry . Figure 10.33 shows two three-dimensional objects that have reflection and/or rotation symmetry. In each case, determine and try to describe the plane(s) of symmetry and the axis (or axes) of symmetry.
eBook - ePubSymmetry Discovered
Concepts and Applications in Nature and Science
- Joe Rosen(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
The intersection of the line and the plane is called the reflection center. The image of a point is therefore located in the plane of the system, directly opposite the initial point with respect to the reflection center, and at the same distance from the reflection center as is the initial point (fig. 3.21). Reflection transformations through different reflection centers do not commute. We do not devote further discussion to this transformation, because it is not really new; a line reflection transformation is completely equivalent to rotation by 180° about the reflection center. Fig. 3.21 Line reflection transformation of 2-dimensional system PROBLEM Convince yourself of this fact. Spatial symmetry Pooh looked at his two paws. He knew that one of them was the right, and he knew that when you had decided which one of them was the right, then the other one was the left, but he never could remember how to begin. (A.A. Milne: The House at Pooh Corner) All the geometric transformations that we considered for 2-dimensional systems are readily applied to 3-dimensional systems. These transformations are displacement, plane and line reflection, glide and rotation. As expected, the addition of a third dimension allows the application of transformations that are inapplicable to lower-dimensional systems. Of these we discuss point reflection and the screw. We also consider the dilation transformation, applicable to systems of any number of dimensions. The action of the displacement transformation on 3-dimensional systems is the obvious generalization of its action on 2-dimensional systems; there is one more dimension to displace into (fig. 3.22). As in the 2-dimensional case, all displacement transformations commute, whether or not they are in the same direction. Fig. 3.22 Some possible displacement transformations for 3-dimensional systems
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Understand and use the concept of a translation. Understand and use the concept of a rotation. 14.2 Reflections (page 541) 21–38 Understand and use the concept of a reflection in a line. Understand and use the definition of a glide reflection. Understand and use the definition of a frieze pattern. 14.3 Dilations and Scale Drawings (page 551) 39–49 Understand and use the concept of dilation. Find the scale factor of a dilation. Use a scale factor to draw a dilation. Use the concept of a dilation to read and create scale drawings. Important Concepts and Formulas Translation A translation is a transformation in which a figure slides but does not turn. Every point of the figure moves the same distance and in the same direction. Rotation A rotation is a transformation in which a figure turns about a point called the center of rotation. The number of degrees a figure rotates is the angle of rotation. Rotational Symmetry A figure has rotational symmetry when a rotation of 180º or less produces an image that fits exactly on the original figure. Reflection A reflection is a transformation in which a figure flips in a line called the line of reflection. A reflection creates a mirror image of the original figure. Glide Reflection A glide reflection is a transformation that is composed of a translation followed by a reflection in a line that is parallel to the direction of the translation. Dilation A dilation is a transformation in which a figure is made larger or smaller with respect to a fixed point called the center of dilation. iStockphoto.com/kali9 Chapter Summary 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.
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 - 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)
As students transition from grades 6–8 into high school, transformations are incorporated not only in the geometry curriculum, but in algebra as well. Transformations include four main types: (1) reflections, (2) translations, (3) rotations, and (4) dilations. Each transformation impacts shapes in different ways. Transformations are used to examine ideas of congruency and similarity as well as scale factor in meaningful ways.Consider the transformations task in figure 5.26 . You may want to make a copy of the diagrams in figure 5.26 before proceeding. For each transformation, use the pre-image and your understanding of the transformation to draw the location of the image.Complete each of the described transformations.Figure 5.26: Reflection, translation, and rotation of a triangle task.How did you determine the locations of the images? What properties are important to consider? With which transformation were you most comfortable? Did any of them create a challenge for you?Visit go.solution-tree.com/mathematics for a free reproducible version of this figure.A reflection is like using a mirror. An object is reflected about a line, and the line becomes an axis of symmetry for the pre-image to the image. Every part of the pre-image (Δ ABC ) on one side of the line of reflection is on the opposite side of the line of reflection in the image (Δ A 'B 'C ') (see figure 5.27 ).What do you notice about the pre-image and the image? Are they congruent or similar? What do you notice about their orientation? In the case of a reflection, the orientation is reversed, but the pre-image and image are congruent shapes. What is the same and what is different between the pre-image and the image? Every point in the image is the same perpendicular distance from
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
In elementary school, translations, rotations, reflections, and glide reflections are frequently called “slides,” “turns,” “flips,” and “slide flips,” respectively. Next, we will see how to apply these transformations to analyze symmetry patterns in the plane. Check for Understanding: Exercise/Problem Set A #1–22 ✔ Symmetry Consider a tessellation of the plane with parallelograms (Figure 16.16). Figure 16.16 We can identify several translations that map the tessellation onto itself. For example, the translations T AB , T CD , and T EF all map the tessellation onto itself. To see that T AB maps the tessellation onto itself, make a tracing of the tessellation, place the tracing on top of the tessellation, and move the tracing according to the translation T AB . The tracing will match up with the original tessellation. (Remember that the tessellation fills the plane.) A figure has translation symmetry if there is a translation that maps the figure onto itself. Every tessellation of the plane with parallelograms like the one in Figure 16.16 has translation symmetry. For the tessellation in Figure 16.16, there are several rotations, through less than 360°, that map the tessellation onto itself. For example, the rotations R A,180° and R G,180° map the tessellation onto itself. Note that G is the intersection of the diagonals of a parallelogram. (Use your tracing paper to see that these rotations map the tessella- tion onto itself.) A figure has rotation symmetry if there is a rotation through an angle greater than 0° and less than 360° that maps the figure onto itself. In Figure 16.17, a tessellation with isosceles triangles and trapezoids is pictured. Again, imagine that the tessellation fills the plane. In Figure 16.17, reflection M l maps the tessellation onto itself, as does M m . A figure has reflection symmetry if there is a reflection that maps the figure onto itself.
eBook - PDFCollege Geometry
A Unified Development
- David C. Kay(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
15. Referring to Problem 14, there exists a unique affine trans-formation f which maps Δ ABC to Δ DEF . Using the coordi-nate form for an affine transformation, use the information given to find the coefficients and thus verify that f is a similitude. 8 .4. . Line.Reflections:.Building.Blocks. for.Isometries.and.Similitudes It is perhaps surprising that all isometries in the plane are simply prod-ucts of reflections in lines—the simplest type of isometry. Line reflections have a multitude of applications in mathematics and physics, as well as in the real world. It is fitting that a section be devoted to this important area of geometry. For convenience, we repeat the definition from a previ-ous section where reflections were originally introduced (in particular, see Section 3.5 ). Definition A reflection in the line l is the mapping which takes each point P in P to the unique point P ′ also in P such that l is the perpen-dicular bisector of the segment PP ′ . (If P ∈ l , we take P ′ = P ; thus l is a line of fixed points for the reflection.) It is evident that one may construct the reflection P ′ for each point P not on l by dropping the perpendicular PM to point M on l , then doubling the segment PM to segment PP ′ (segment-doubling theorem). It should be noticed that reflections are special cases of affine reflections. It is almost obvious that reflections preserve distance, and consequently, reflections are isometries. In Figure 8.21 is shown two cases, (1) when A and B lie on the same side of l , and (2) when they do not. Thus, we see that in case (1) ◊ ABNM ≅ ◊ A ′ B ′ NM (by SASAS), so that AB = A ′ B ′ by CPCPC, and in case (2) AB ′ = A ′ B by case (1) so that trapezoid AA ′ BB ′ is isosceles, with congruent diagonals, AB and A ′ B ′ . We include one property of reflections that has practical applications.
eBook - PDFIndra's Pearls
The Vision of Felix Klein
- David Mumford, Caroline Series, David Wright(Authors)
- 2002(Publication Date)
- Cambridge University Press(Publisher)
As in most of mathematics, these ordinary words are being used in specialised and very precise ways. 10 The language of symmetry We had better explain carefully what is meant, because without using them it would be virtually impossible to write this book. In its widest sense, a transformation of the plane means simply a rule which assigns to each point P in the plane a new point Q. The rule might be: ‘the new point is 3 inches to the left of the old one’, or ‘the new position is obtained by rotating 90 ◦ with respect to the centre point O’. The new point Q we get to is called the image 1 of the starting point P . 1 Mathematicians have a predilection for giving special names to everything they talk about. They don’t do this just to sound imposing: think of it like a surgeon laying out her instruments and checking everyone knows their correct names, to make sure she gets the right thing when she calls to the nurse. Although a tremendous number of rules can be thought up, we shall only be thinking about rules whose effect can be undone. For example, the effect of the translation ‘move 3 inches to the left’, can be undone by the rule ‘move 3 inches to the right’, see Figure 1.7. Figure 1.7. Dr. Stickler being transported by the transformation T which moves him 3 inches to the right. In the second frame you can see him transported by the inverse transformation T −1 , which moves him back 3 inches to the left. T T −1 A more concrete way to think of a transformation is to consider it as a procedure for physically moving the points of the plane to new locations. During the motion, the relative positions of points may get hugely distorted, but whatever figures or objects are in the plane get carried along and move to their new positions at the same time. You can see the effect of such a distortion on the classical bust of Paolina shown on the left in Figure 1.8.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
CHAPTER 16 Geometry LEARNING OUTCOMES 16.1 Introducing geometry and the attributes of two-dimensional shapes and three-dimensional objects. 16.2 Describing the roles that location, position and spatial relationships play in geometry. 16.3 Identifying applications of geometric transformations. 16.4 Understanding how exploration with geometry can help build children’s spatial reasoning and visualisation skills. ‘Inspiration is needed in geometry, just as much as in poetry.’ Aleksandr Pushkin (Russian writer, 1799–1837) Pdf_Folio:540 Chapter 16 concept map Geometry Geometric reasoning and visualisation Transformation and symmetry • Translation (slide) • Rotation (turn) and refection (fip) • Enlargement 2D shapes Recognition, description and knowledge of properties and relationships 3D objects Prisms, pyramids, circular solids and polyhedra Recognition, description and knowledge of properties and relationships Location and position Movement and arrangement of objects, pathways Maps and grids (including scale and plans) Introduction Over time, perspectives about the importance of geometry have shifted away from focusing on definitions and proofs towards the study of geometry as essential to mathematics learning at all levels, from prior to school through to secondary and beyond. Geometry is closely linked to measurement, number and patterns. Furthermore, spatial reasoning is now seen as fundamental in many careers and applications outside of mathematics, or connected with maths, particularly STEM fields. In previous curricula, a lack of understanding and connection to the real world led to negative views of geometry. These views were based on a number of factors: insufficient time for children to play with concrete materials in their early years; inappropriate curriculum materials for primary children which focused on an abstract approach; and children’s limited exposure to geometric ideas due to an overemphasis on number.
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