Mathematics
Translations
In mathematics, translations refer to the movement of a geometric figure from one location to another without changing its size, shape, or orientation. This movement is typically described using coordinate notation, where each point of the figure is shifted by a specified amount in a specified direction. Translations are a fundamental concept in geometry and are essential for understanding transformations in mathematics.
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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)
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. To convince your students that there are no reflections, make a copy of the print, cut out one of the lizards, and reflect it. The reflected image does not occur in the print. Reflection Classroom Tip 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. 532 Chapter 14 Transformations Translations Definition of 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. Slide Slide The original figure and its image are congruent. EXAMPLE 2 Identifying Translations Determine whether the blue figure is a translation of the red figure. a. b. SOLUTION a. The red trapezoid slides to form the blue trapezoid. So, the blue trapezoid is a translation of the red trapezoid. b. The red 3 turns to form the blue 3. So, the blue 3 is not a translation of the red 3. EXAMPLE 3 Identifying Translations in Real Life You rearrange the icons on your smartphone. Do the icons represent examples of Translations? Explain your reasoning. Original Rearranged SOLUTION. Except for the message icon ( ) , all of the icons slide without changing their size or shape.
eBook - PDFBackgrounds Of Arithmetic And Geometry: An Introduction
An Introduction
- Dan Branzei, Radu Miron(Authors)
- 1995(Publication Date)
- World Scientific(Publisher)
Fig.23 150 Backgrounds of Arithmeiic and Geometry. An Introduction Theorem 5.1. Any translation is an isometry; the inverse of a translation is c translation; the composition of two Translations of vectors u,v,is the translation oj vector u+v Theorem 5.2. A translation T of non-null vector u does not have fixed points, preserves the straight lines parallel with u, planes a parallel with u and does not preserve circles or spheres. Theorem 53. The composition S 2 oS x of two symmetries related to the parallel axes dy , d 2 is a translation of vector u perpendicular on d x , having a size equal to the double of the distance between d x and d 2 and the sense from d to d 2 . Any translation is decomposed into product of symmetries. Proof. At the beginning, we examine the case when S is a plane. Let us consider OO x 1 d x , O e d x and an axis Ox oriented from O to O x e d 2 . Let also 0 2 be symmetrical to O related to O x . Let us suppose O x (a, O). Point A(x, y) passes through symmetry S x i nt o Ax y where, obviously, * ' = -xy'=-y'. Points xy) passes through symmetry S 2 moA (x,y), where x= la - x'= la + x a n d / ' = y-= y. Transformation T = S 2 ° 5, leads A(x, y) into A (x + 2a, y), so it is a translation of vector -> -> AA = OQ 2 . If 5 coincides with E 3 , we shall consider for the arbitrary point A the plane a, incident with A and perpendicular on d x Let U, U x be the points in which d x , d 2 pricks a. With the above same notations it is easy to find out that UU X is a median line in the -* -> -> triangle A'AA and therefore AA = 2-UU^ The vector UU X does not depend on point A so the conclusion of the theorem is proved A simple analytical expression of transformation T= S 2 oS 1 is obtained taking d x as axis Oz (so d x has equations x = y = 0) and d 2 included into the plane xOz (d 2 of equations x - a = 0 = y). Then, for A {x, y, z point T (A) has the co-ordinates is /
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