Using Similar Polygons

11 Key excerpts on "Using Similar Polygons"

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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)

  Section 13.2 Similarity 509 Similar Polygons Definition of Similar Polygons Two polygons are similar polygons when their corresponding side lengths are proportional and their corresponding angles are congruent. EXAMPLE 9 Determining Whether Two Polygons are Similar Is ABCD similar to WXYZ? Explain your reasoning. A W X Y Z D C B 8 80 40 150 110 4 15 11 120° 90° 105° 105° 45° 90° SOLUTION Recall from Section 10.2 that the sum of the angle measures of a quadrilateral is 360º. Using this fact, you can determine that the missing angle measures are 45º for angle B and 120º for angle Z. So, the corresponding angles of the quadrilaterals are congruent. Also, each side length of the larger quadrilateral is 10 times longer than the corresponding side length of the smaller quadrilateral. So, the side lengths are proportional. Because the corresponding angles are congruent and the corresponding side lengths are proportional, ABCD ~ WXYZ. EXAMPLE 10 Comparing Constellations Are the quadrilateral Ursa Major Ursa Minor Polaris, North Star Merak Dubhe portions of the Big Dipper and the Little Dipper constellations similar? Explain your reasoning. SOLUTION Trace the quadrilateral portion of each constellation in the drawing and orient them so that corresponding sides align as shown. From this, you can see that Ursa Major Ursa Minor the corresponding angles of the quadrilaterals are not congruent. So, the quadrilateral portions of the constellations are not similar. Mathematical Practices Reason Abstractly and Quantitatively MP2 Mathematically proficient students make sense of quantities and their relationships in problem situations. 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.

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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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  Introductory Technical Mathematics

  • John Peterson, Robert Smith(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Often, scale drawings are in the form of similar polygons or com-binations of similar polygons. Similar polygons have the same number of sides, equal corresponding angles, and proportional corresponding sides. The symbol , means similar. Corresponding angles of similar polygons are equal. Corresponding sides of similar polygons are proportional. EXAMPLES 1. Refer to similar polygons in Figure 23–11. Figure 23–11 B A E C F D B 9 A 9 E 9 C 9 F 9 D 9 Given: ABCDEF , A 9 B 9 C 9 D 9 E 9 F 9 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. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Unit 23 CONGRUENT AND SIMILAR FIGURES 585 Conclusion: The corresponding angles are equal. / A 5 / A 9 / D 5 / D 9 / B 5 / B 9 / E 5 / E 9 / C 5 / C 9 / F 5 / F 9 The corresponding sides are proportional. AB A 9 B 9 5 BC B 9 C 9 5 CD C 9 D 9 5 DE D 9 E 9 5 EF E 9 F 9 5 FA F 9 A 9 2. Refer to the similar polygons in Figure 23–11. If AB 5 9.1 cm, A 9 B 9 5 10.5 cm, and EF 5 5.2 cm, what is the length of E 9 F 9 ? We have AB A 9 B 9 5 EF E 9 F 9 . Substitute the given data 9.1 cm 10.5 cm 5 5.2 cm E 9 F 9 E 9 F 9 5 s 5.2 cm ds 10.5 cm d 9.1 cm 5 6.0 cm Ans Perimeters of Polygons The distance around a polygon, or the sum of all sides, is called the perimeter ( P ) . In Figure 23–11: P 5 AB 1 BC 1 CD 1 DE 1 EF 1 FA P 9 5 A 9 B 9 1 B 9 C 9 1 C 9 D 9 1 D 9 E 9 1 E 9 F 9 1 F 9 A 9 The perimeters of two similar polygons have the same ratio as any two correspond-ing sides.

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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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  Elementary Geometry for College Students

  • Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  We will consider cylinders in greater detail in Chapter 9. KEY CONCEPTS Similar Polygons Congruent Polygons Corresponding Vertices, Angles, and Sides 5.2 Similar Polygons Two congruent polygons are also similar polygons. Figure 5.3 (a) A C B (b) D F E Exercises 42, 43 230 CHAPTER 5 ■ SIMILAR TRIANGLES O . J . 6 o u n c e s O . J . 6 o u n c e s O . J . 1 6 o u n c e s O . J . 1 6 o u n c e s T U N A T U N A (a) (b) (c) 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 congruent 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 uni2220A in one polygon is congruent to uni2220H in the second polygon, then vertex A corresponds to vertex H, and this is symbolized A 4 H; we can indicate that uni2220A corresponds to uni2220H by writing uni2220A 4 uni2220H. A pair of angles like uni2220A and uni2220H are corresponding angles, and the sides determined by consecutive and corresponding vertices are corresponding sides of the similar polygons. For instance, if A 4 H and B 4 J , then AB corresponds to HJ . EXAMPLE 1 Given similar quadrilaterals ABCD and HJKL with congruent angles as indicated in Figure 5.5, name the vertices, angles, and sides that correspond to each other. (a) B C A D (b) H L J K Figure 5.5 SOLUTION Because uni2220A _ uni2220H, it follows that A 4 H and uni2220A 4 uni2220H. Similarly, B 4 J and uni2220B 4 uni2220J C 4 K and uni2220C 4 uni2220K D 4 L and uni2220D 4 uni2220L Associating pairs of consecutive and corresponding vertices of similar polygons, we determine the endpoints of the corresponding sides.

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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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  Prealgebra

  • 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 5 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  ^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  ^CDE , a correspondence between the vertices of the triangles is established.       ^PQR  ^CDE Since the triangles are similar, corresponding angles are congruent: P > C Q > D R > E The lengths of the corresponding sides are proportional. To simplify the notation, we will now let PQ 5 m(PQ ) , CD 5 m(CD ) , QR 5 m(QR ) , and so on. PQ CD 5 QR DE QR DE 5 PR CE PQ CD 5 PR CE Written in a more compact way, we have PQ CD 5 QR DE 5 PR CE EXAMPLE 4 Self Check 4 If ^GEF  ^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. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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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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  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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  Transformational Plane Geometry

  • Ronald N. Umble, Zhigang Han(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 8 Similarity Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, trans-lations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Common Core State Standards for Mathematics CCSS.MATH.CONTENT.8.G.A.4 In this chapter we consider transformations that magnify or stretch the plane. Such transformations are called “similarities” or “size transformations.” One uses a similarity to relate two similar triangles in much the same way one uses an isometry to relate two congruent triangles. The “stretch” similarities, which linearly expand the plane radially outward from some fixed point, are especially important because they provide an essential component of every non-isometric similarity. Indeed, we shall prove that every similarity is one of the following four distinct similarity types: an isometry, a stretch, a stretch rotation, or a stretch reflection. 8.1 Plane Similarities Recall Definition 93: A similarity of ratio r > 0 is a transformation α : R 2 → R 2 such that if P and Q are points, P 0 = α ( P ) , and Q 0 = α ( Q ) , then P 0 Q 0 = rPQ. Note that if r = 1 , then P 0 Q 0 = PQ and α is an isometry. Furthermore, if α has distinct fixed points P and Q, then P 0 = P, Q 0 = Q, and P 0 Q 0 = P Q, the ratio of similarity r = 1 and α is an isometry. Thus α is the identity or a reflection by Theorem 170. Of course, if α has three non-collinear fixed points, then α = ι by Theorem 135. This proves: Proposition 254 A similarity of ratio 1 is an isometry; a similarity with two or more distinct fixed points is a reflection or the identity; a similarity with three non-collinear fixed points is the identity. 169 170 Transformational Plane Geometry Similarities are bijective by Proposition 95. In fact: Proposition 255 The set S of all similarities is a group with respect to com-positions.

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  Solving Problems in Geometry

  Insights and Strategies for Mathematical Olympiad and Competitions

  • Kim Hoo Hang, Haibin Wang;;;(Authors)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2

  Similar Triangles

  Similar triangles are the natural extension of the study on congruent triangles. While congruent triangles describe a pair of triangles with identical shape and size (area), similar triangles focus on the shape. The diagram below gives an illustration.

  Indeed, similar triangles are even more powerful tools than congruent triangles. Many interesting properties and important theorems in geometry could be proved by similar triangles.

  One would see in this chapter that the Intercept Theorem plays a fundamental role in studying similar triangles, while the proof of this theorem is based on an even more fundamental concept: area. 2.1 Area of a Triangle

  It is widely known that the area of ΔABC, denoted by [ΔABC] or S

  ΔABC

  , is given by where h denotes the height on BC.

  Of course, one may replace BC and h by any side of the triangle and the corresponding height on that side.

  Notice that implies that if two triangles have equal bases and heights, they must have the same area. Even though this is a simple conclusion, it has a number of (important) variations:

  • In a trapezium ABCD where AD // BC and AC,BD intersect at E, we have [ΔABC] = [ΔDBC] because both triangles have a common base and equal heights.

  By substracting [ΔBCE] on both sides of the equation, we have [ΔABE]=[ΔCDE]. Refer to the diagram above.

  • In a triangle ΔABC where M is the midpoint of BC, we must have [ΔABM]=[ΔACM]. Let D be any point on AM. We also have [ΔBDM]=[ΔCDM]. Refer to the diagram below.

  It follows that [ΔABD]=[ΔACD]. Since ΔABD and ΔACD have a common base AD, we conclude that the perpendicular distance from B,C respectively to the line AM is the same.

  Notice that the conclusion above still holds even if D is a point on AM extended. Refer to the diagram below.

  If M is the midpoint of BC, can you see [ΔABD]=[ΔACD]?

  • Refer to the diagram below. Given a triangle ΔABC, extend BC to D such that BC = CD. E is a point on AC. Draw a parallelogram CDFE. Connect BE, BF and AF. One sees that the area of the shaded region is equal to the area of ΔABC

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