8.5 Transformational Geometry, Symmetry, and Tessellations 509 In Example 5 part b), note that other than the vertex labels, the positions of the parallelogram before and after the rotation are identical. We will revisit rotations like this when we discuss rotational symmetry later in this section. Now we discuss the fourth rigid motion, glide reflection. Glide Reflections Definition: Glide Reflection A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection. In a glide reflection, the reflection line and the translation vector must be parallel to each other. Consider triangle ABC (shown in blue), translation vector v, and reflection line l in Fig. 8.69. Note that the translation vector v and reflection line l are parallel to each other. The translation of triangle ABC using translation vector v is triangle A BC ′ ′ ′ (shown in red). The reflection of triangle A BC ′ ′ ′ about reflection line l is triangle A B C ″ ″ ″ (shown in green). Thus, triangle A B C ″ ″ ″ is the glide reflection of triangle ABC, using translation vector v and reflection line l. Notice from Fig. 8.69 that had we performed the reflection first, followed by the translation, the triangle A B C ″ ″ ″ would still end up in the same final position. In a glide reflection, the order in which the translation and the reflection are performed does not matter. l v A0 A9 A B0 B9 B C9 C C0 Figure 8.69 Example 6 A Glide Reflection of a Trapezoid Construct a glide reflection of trapezoid ABCD, given in Fig. 8.70, using translation vector v and reflection line l. l v B A C D Figure 8.70
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