Similarity

9 Key excerpts on "Similarity"

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  Teaching and Learning Geometry

  • Doug French(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Finally it is an interesting exercise to make some square-based pyramids with equilateral triangles as faces and use them, together with some tetrahedra, to make a square-based pyramid which is 'twice as big'. This will require the equivalent of sixteen tetrahedra. CONCLUSION Similarity is one of the big important ideas in geometry with a wealth of applications to real situations as well as being a frequent feature of theorems, proofs and problems. Its direct link to the transformation of enlargement provides an alternative approach involving scale factors, which is a simpler idea to grasp initially than the more traditional approach through equal ratios. It also extends readily to problems involving areas and volumes in similar shapes and solids. As with all areas of geometry new ideas should be accompanied by practical tasks. Accurate drawing, working with actual shapes and solids, making models and using dynamic geometry software in a variety of ways all have a valuable part to play in developing the essential intuitive feel for the ideas of enlargement and Similarity. The word similar has an everyday meaning which implies a certain sameness. This is a potential source of confusion because it is less precise than the mathematical meaning. In everyday language we may say that some shapes, for example a set of rectangles, are similar or the 'same shape' because they all have four sides and right angles. Mathematical Similarity certainly requires figures to have the same number of sides and identical angles, but it also requires corresponding lengths to be in the same proportion. Proportionality is the key property of Similarity and it is an idea that is a considerable source of difficulty. That is a major reason for linking Similarity to enlargement and for introducing scale factors because they offer a more accessible approach than equal ratios.

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  First Course in Euclidean Geometry, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. One can be obtained from the other by uniformly stretching the same amount on all directions, possibly with additional rotation and reflection, i.e., both have the same shape, or one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Here we assumes that a scaling, enlargement or stretch can have a scale factor of 1, so that all congruent shapes are also similar, but some school text books specifically exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different to qualify as similar. Similar triangles To understand the concept of Similarity of triangles, one must think of two different concepts. On the one hand there is the concept of shape and on the other hand there is the concept of scale. ________________________ WORLD TECHNOLOGIES ________________________ If you were to draw a map, you would probably try to preserve the shape of what you are mapping, while you would make your picture at a unit rate that is in proportion to the original size or value. In particular, similar triangles are triangles that have the same shape and are up to scale of one another. For a triangle, the shape is determined by its angles, so the statement that two triangles have the same shape simply means that there is a correspondence between angles that preserve their measures. Formally speaking, we say that two triangles and are similar if either of the following conditions holds: 1.

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  Mathematical Practices, Mathematics for Teachers

  Activities, Models, and Real-Life Examples

  • Ron Larson, Robyn Silbey(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Describe how the definition of similar differs when used in mathematics and when used in real life. Give an example of how two objects might be similar in real life but not in mathematics. 22. Writing Identify two careers that use similar figures. Explain how similar figures are used. 23. Grading Student Work On a diagnostic test, one of your students does the following work. Explain what the student did wrong. Which topics would you encourage the student to review? The two triangles are similar because they are both right triangles. 24. Grading Student Work On a diagnostic test, one of your students does the following work. Explain what the student did wrong. Which topics would you encourage the student to review? The two triangles are similar because of the SSS Similarity Theorem. 6 3 8 6 4 3 25. Activity: In Your Classroom Cut an isosceles right triangle with legs at least one foot long out of cardboard. Put the triangle on the floor in front of a desk and use a string to align the hypotenuse of the triangle to the top of the desk, as shown in the figure. 45° A B C a. Measure the legs of △ABC. b. Compare the ratio of the legs of the cardboard triangle with the ratio of the legs of △ABC. c. Describe the concepts your students will practice and discover by doing this activity. 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 13.2 Similarity 513 26. Activity: In Your Classroom To find the height of your classroom wall using the triangle and desk in Exercise 25, put the triangle on the desk.

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  The Space of Mathematics

  Philosophical, Epistemological, and Historical Explorations

  • Javier Echeverria, Andoni Ibarra, Thomas Mormann, Javier Echeverria, Andoni Ibarra, Thomas Mormann(Authors)
  • 2012(Publication Date)
  • De Gruyter

    (Publisher)

  Notice, however, that the Similarity often does not go too far. For example, there are no obvious structural similarities between basic types of algebraic expressions and fundamental types of geometrical curve; hence classification has been a difficult task for algebraic geometry (or should it sometimes be thought of as geometrical algebra?) from Newton onwards. Again, dis-Similarity is evident in problems involving the roots of equations, where the algebra is unproblematic but a root does not lead to a geometrically intelligible situation (an area becomes negative, say). The occurrence of complex zeroes in a polynomial can be still more problematic as geometry, since their presence is not reflected in the geometrical representation of the Structure-Similarity as a Cornerstone of Philosophy of Mathematics 95 corresponding function in the real-variable plane, and a pair of complex-variable planes will represent the argument and the value of the function but necessarily do not reflect the function itself. Other cases includes a situation where the algebraic solution of a geometrical problem supplies a circle (say) as the required locus whereas only an arc of it actually pertains to the problem. In an undeservedly forgotten examination of 'the origin and the limits of the correspondence' between these two branches of mathematics. [Coumot 1847] explores in a systematic way these and other cases and sources of such structural non-Similarity. The mathematics is quite elementary; the philosophy is far from trivial. 2.2. Representations and Icons Sometimes these intra-mathematical structure-similarities are put forward more formally as representations, such as in the geometrical characterisation of complex numbers in the plane, or Cauchy's definition by such means of infi-nitesimals in terms of sequences of real values passing to zero.

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  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  Geometry will draw the soul toward truth.

  PLATO

  CHAPTER SIX

  Similarity

  6.1  INTRODUCTION

       In the sixth century B.C ., the illustrious mathematician Pythagoras founded a school which adopted the philosophical premise that the examination of everything in nature, geometric forms included, would yield properties that could be expressed in terms of whole numbers. According to legend, the Pythagoreans were onboard ship when one of them demonstrated to the others that, no matter what unit of measure was employed, it was impossible to express the lengths of the legs and hypotenuse of an isosceles right triangle as whole numbers. Since this fact violated Pythagorean doctrine, it was viewed as a serious threat by the Pythagoreans, who responded by throwing its heretical author overboard–or so the story goes.

       Thus was generated a “crisis” in mathematics which lasted for nearly two hundred years, until Eudoxus invented his elegant and ingenious theory of ratio and proportion. Eudoxus’ work is generally believed to have furnished the entire content of Book V of Euclid’s Elements.

       In previous chapters, we examined the congruence relation between geometric sets of points. Specifically, we defined congruence for segments, angles, and triangles. When we say that two such figures are congruent, we are saying that they are the same shape and the same size.

       In this chapter we will begin to consider relations between figures that are the same shape but not necessarily the same size. We have a name for such figures: We say they are similar.

       Eudoxus’ theory of proportion provides the tools we will use to explore and refine the Similarity relation. The notion of Similarity will then lead us to a proof of what is known as the Pythagorean Theorem

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  Basic Mathematics for College Students with Early Integers

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The formal definition of similar triangles requires that we establish a correspondence between the vertices of the triangles. The definition also involves the word proportional. Recall that a proportion is a mathematical statement that two ratios (fractions) are equal. An example of a proportion is 1 2 H11005 4 8 In this case, we say that 1 2 and 4 8 are proportional. Similar Triangles Two triangles are similar if and only if their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are proportional. Refer to the figure below. If ^PQR H11011 ^CDE , name the congruent angles and the sides that are proportional. Q P R C D E Strategy We will establish the correspondence between the vertices of ^PQR and the vertices of ^CDE . WHY This will, in turn, establish a correspondence between the congruent corresponding angles and proportional sides of the triangles. Solution When we write ^PQR H11011 ^CDE , a correspondence between the vertices of the triangles is established.       ^PQR H11011 ^CDE Since the triangles are similar, corresponding angles are congruent: H11028P H11061 H11028C H11028Q H11061 H11028D H11028R H11061 H11028E The lengths of the corresponding sides are proportional. To simplify the notation, we will now let PQ H11005 m(PQ ) , CD H11005 m(CD ) , QR H11005 m(QR ) , and so on. PQ CD H11005 QR DE QR DE H11005 PR CE PQ CD H11005 PR CE Written in a more compact way, we have PQ CD H11005 QR DE H11005 PR CE EXAMPLE 4 Self Check 4 If ^GEF H11011 ^IJH , name the congruent angles and the sides that are proportional. F G E J I H Now Try Problem 39 Copyright 2019 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.

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  Geometry Transformed

  Euclidean Plane Geometry Based on Rigid Motions

  • James R. King(Author)
  • 2021(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 8 Dilations and Similarity In this chapter, we will begin the study of similar figures. Similarity is a relationship among figures much like congruence, except that instead of “same size, same shape,” it formalizes an idea of “enlarged or reduced size, same shape”. As with congruence, certain transformations will be used to define this relation; these transformations are called similitudes. Definition 8.1. A similitude with scale factor ? > 0 is a transformation that scales distances by the factor ? . To be specific, if ? and ? are any points and ? is a similitude, then ‖?(?)?(?)‖ = ?‖??‖ . If ? is a similitude, ? −1 is a similitude with scale factor 1/? . To see ‖? −1 (?)? −1 (?)‖ = (1/?)‖??‖ for all ? and ? , let ? = ? −1 (?) and ? = ? −1 (?) . Then this equation becomes ‖??‖ = (1/?)‖?(?)?(?)‖ , which is true for all ? and ? . By the Dilation Axiom, any dilation 𝒟 ?,? is a similitude with scale factor ? . From its definition, a rigid motion is a similitude with scale factor 1 . Conversely, a similitude with scale factor ? = 1 is an isometry and so by Theorem 4.10 is a rigid motion. If ? and ? are similitudes with scale factors ? and ? , then ?? is a similitude with scale factor ?? , since ‖??(?)??(?)‖ = ?‖?(?)?(?)‖ = ??‖??‖ for any points ? and ? . Similitudes preserve angle measure. This is not part of the definition, but we can prove this as a theorem. Theorem 8.2 (Similitudes and Angles) . Every similitude preserves angle measure. Con-sequently, since angle measures of 0 and 180 are preserved, a similitude maps lines to lines, segments to segments, and rays to rays. Proof. Let ? be a similitude with scale factor ? . For any point ? , 𝒟 ?,1/? is a similitude with scale factor 1/? . Then, as observed above for ?? , the product ? = 𝒟 ?,1/? ? is a similitude with scale factor ?(1/?) = 1 , so it is an isometry and hence is a rigid motion by Theorem 4.10.

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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Note that the symbol for con-gruence combines the symbols for Similarity and equality; that is, congruent polygons always have the same shape and the measures of corresponding parts are equal. ( ) ( ) ( ) . Similar Polygons Congruent Polygons Corresponding Vertices, Angles, and Sides KEY CONCEPTS Similar Polygons 5.2 (a) A C B (b) D F E Figure 5.3 Two congruent polygons are also similar polygons. While two-dimensional figures such as and in Figure 5.3 can be sim-ilar, it is also possible for three-dimensional figures to be similar. Similar orange juice con-tainers are shown in Figures 5.4(a) and 5.4(b) on page 218. Informally, two figures are “similar” if one is an enlargement of the other. Thus, a tuna fish can and an orange juice can are not similar, even if both are right-circular cylinders [see Figures 5.4(b) and 5.4(c) on page 218.] We will consider cylinders in greater detail in Chapter 9. DEF ABC Copyright 2013 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. Figure 5.4 In this chapter, the discussion of Similarity will generally be limited to plane figures. For two polygons to be similar, it is necessary that each angle of one polygon be con-gruent to the corresponding angle of the other. However, the congruence of angles is not sufficient to establish the Similarity of polygons. The vertices of the congruent angles are corresponding vertices of the similar polygons. Consider Figure 5.5. If in one poly-gon is congruent to in the second polygon, then vertex A corresponds to vertex H , and this is symbolized ; we can indicate that corresponds to by writing .

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  Euclidean Geometry and Transformations

  • Clayton W. Dodge(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  similitude is also used for this mapping.

  This section completes our study of similarities and their properties, so that we may proceed rapidly to the applications in Sections 27 to 29 . Although the theory we develop here is not as extensive as that for isometries in Chapter 2 , it is quite sufficient for our purposes.

  26.2 Theorem A Similarity of ratio 1 is an isometry.

  26.3 Theorem A Similarity maps segments into segments.

  Let a Similarity of ratio k map A to A′ and B to B′. Take any point P between A and B and let the image of P be P′. Then

  so P′ lies on segment A′B′.

  26.4 Theorem A Similarity is a transformation of the plane.

  26.5 Theorem A Similarity preserves angles.

  Mark points B and C on the two sides of angle A to form a triangle ABC. Let the Similarity map triangle ABC to triangle A′B′C′ by Theorem 26.3 . Then

  Thus triangles ABC and A′B′C′ are similar by SSS. Hence and the theorem follows.

  26.6 Corollary A Similarity maps a triangle into a similar triangle. More generally, it maps a polygon into a similar polygon.

  26.7 Theorem A Similarity is determined by any three noncollinear points and their images.

  26.8 Theorem There are exactly two similarities, one direct and one opposite, that map any point pair A, B into any other point pair A′, B′ (understanding that A ≠ B, A′ ≠ B, A′ is the image of A, and B′ is the image of B).

  26.9 Theorem Each direct Similarity of ratio k that is not an isometry is the product of a rotation and a homothety of ratio k having the same center. Furthermore, such a product is commutative.

  Clearly such a product is a direct Similarity. By Theorem 26.8 , this Similarity is determined by a segment AB and its image A′B′. Let AA′ and BB′ meet at Q, and draw the circles through A, B, Q, and through A′, B′, Q to meet again at O, as shown in Fig. 26.9 . Then we have

  and

  by the properties of angles inscribed in a circle. Hence triangles ABO and A′B′O are similar. Now, if AB is rotated about O through angle AOA′ to A1 B1 then A1 B1 is parallel to A′B′ and triangles OA1 B1 and O, A′B′ are similar, so B1 lies on line OB′. Thus the homothety H(O, k) maps A1 Bl to A′B′. This rotation and homothety satisfy the theorem. Now, if the figure formed by O, A1 , B1 , A′, B′ is rotated about O through angle A

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