8.5 Transformational Geometry, Symmetry, and Tessellations 511 of the figure before the rigid motion, then the rigid motion is a symmetry and we say that the figure has symmetry. For a two-dimensional figure, there are four types of symmetries: reflective symmetry, rotational symmetry, translational symmetry, and glide reflective symmetry. In this textbook, however, we will discuss only reflective symmetry and rotational symmetry. Consider the polygon and reflection line l shown in Fig. 8.72(a). If we use the rigid motion of reflection and reflect the polygon ABCDEFGH about line l, we get polygon A BC D E F G H ′ ′ ′ ′ ′ ′ ′ ′ shown in Fig. 8.72(b)). Compare Fig. 8.72(a) with Fig. 8.72(b). Although the vertex labels are different, the reflected polygon is in the same position as the polygon in the original position. Thus, we say that the polygon has reflective symmetry about line l. We refer to line l as a line of symmetry. Notice the line of symmetry is also the reflection line. A9 B9 H9 G9 C9 F9 D9 E9 l l A H B C G F D E (a) (b) Figure 8.72 Recall Example 2, in which hexagon ABCDEF was reflected about reflection line l. Examine the hexagon in the original position (Fig. 8.51) and the hexagon in the final position after being reflected about line l (Fig. 8.52). Other than the labels of the vertices, the beginning and ending positions of the hexagon are identical. Therefore, hexagon ABCDEF has reflective symmetry about line l. Learning Catalytics Keyword: Angel-SOM-8.5 (See Preface for additional details.) Example 7 Reflective Symmetries of Polygons Determine whether the polygon shown in Fig. 8.73 has reflective symmetry about each of the following lines. a) Line l b) Line m Solution a) Examine the reflection of the polygon about line l as seen in Fig. 8.74(a). Notice that other than the vertex labels, the beginning and ending positions of the polygon are identical. Thus, the polygon has reflective symmetry about line l. b) Examine the reflection of the polygon about line m as seen in Fig. 8.74(b). Notice that the position of the reflected polygon is different from the original position of the polygon. Thus, the polygon does not have reflective symmetry about line m. A H G m F E D B C l Figure 8.73
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