510 CHAPTER 8 Geometry Symmetry We are now ready to discuss symmetry. Our discussion of symmetry involves a rigid motion of an object. Solution To construct the glide reflection of trapezoid ABCD, first translate the trapezoid 5 units to the right, as indicated by translation vector v . The translated trapezoid is labeled A BC D, ′ ′ ′ ′ shown in red in Fig. 8.71(a). Next, reflect trapezoid A BC D ′ ′ ′ ′ about reflection line l. This reflected trapezoid, labeled A B C D , ″ ″ ″ ″ is shown in green in Fig. 8.71(b). The glide reflection of trapezoid ABCD is trapezoid A B C D . ″ ″ ″ ″ l v B (a) A C D B9 A9 C9 D9 Figure 8.71 l v B A C D B9 A9 C9 D9 D0 C0 A0 B0 (b) 7 Now try Exercise 33 TECHNOLOGY TIP There’s a Geometry App for That! Many of the concepts that we discuss throughout this section and throughout this entire chapter can be further explored with the use of an app on your smartphone or tablet computer. Several of these apps allow users to construct both two- and three-dimensional objects. Users are then allowed to explore these figures further, including calculating area and perimeter of polygons, area and circumference of circles, and volume and surface area of three-dimensional figures. Several apps also allow users to explore the transformational geometry that we are studying in this section. Once a two-dimensional figure is constructed, the user is then able to perform reflections, translations, rotations, and glide reflections on the figure. As always, consult with your instructor before using these apps when completing work related to your course. RECREATIONAL MATH Symmetry in Nature Symmetry can be found everywhere in nature. One type of symmetry in nature is reflective symmetry, or bilateral symmetry. For example, if you draw a line down the center of a maple leaf, you will often find that one half has the same shape as the other half. Rotational symmetry, or radial symmetry , is also found in nature. Starfish, sand dollars, and many flowers all display rotational symmetry. For example, if you rotate a daisy ° ° 90,180, or ° 270, the rotated daisy will look identical to the original daisy. There are many other examples of symmetry in nature. In Exercise 56 you are asked to find other examples of reflective symmetry and rotational symmetry. Mickes Photos/Shutterstock Definition: Symmetry A symmetry of a geometric figure is a rigid motion that moves the figure back onto itself. That is, the beginning position and ending position of the figure must be identical. Suppose we start with a figure in a specific position and perform a rigid motion on this figure. If the position of the figure after the rigid motion is identical to the position Serp/Shutterstock
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