512 CHAPTER 8 Geometry We will now discuss a second type of symmetry, rotational symmetry. Recall that a positive degree measure indicates that the rotation is counterclockwise. Consider the polygon and rotation point P shown in Fig. 8.75(a). The rigid motion of rotation of polygon ABCDEFGH through a 90° angle about point P gives polygon A BC D E F G H ′ ′ ′ ′ ′ ′ ′ ′ shown in Fig. 8.75(b). Compare Fig. 8.75(a) with Fig. 8.75(b). Although the vertex labels are different, the position of the polygon before and after the rotation is identical. Thus, we say that the polygon has 90° rotational symmetry about point P. We refer to point P as the point of symmetry. C9 B9 D9 E9 A9 F9 H9 G9 A H P P B C G F D E (a) (b) Figure 8.75 Recall Example 5, in which parallelogram ABCD was rotated 90° about point P in part (a) and 180° in part (b). First examine the parallelogram in the original position in Fig. 8.66 and the 90° rotated parallelogram in Fig. 8.67. Notice that the position of the parallelogram after the 90° rotation is different than the original position of the parallelogram. Therefore, parallelogram ABCD in Fig. 8.66 does not have 90° rotational symmetry about point P. Next, examine the 180° rotated parallelogram in Fig. 8.68. Notice that other than the vertex labels, the positions of the two parallelograms ABCD and A BC D ′ ′ ′ ′ are identical with respect to rotation about point P. Therefore, parallelogram ABCD in Fig. 8.66 has 180° rotational symmetry about point P. (a) F9 G9 H9 A9 B9 C9 E9 D9 l Figure 8.74 H9 G9 F9 E9 B9 A9 C9 D9 (b) m 7 Now try Exercise 41 Example 8 Rotational Symmetries Determine whether the polygon shown in Fig. 8.76 has rotational symmetry about point P for rotations through each of the following angles. a) 90° b) 180° Solution a) To determine whether the polygon has 90° rotational symmetry about point P, we rotate the polygon 90° as shown in Fig. 8.77(a). Compare Fig. 8.77(a) with Fig. 8.76. Notice that the position of the polygon after the rotation in Fig. 8.77(a) is different than the original position of the polygon (Fig. 8.76). Therefore, the polygon does not have 90° rotational symmetry about point P. N O Q M A B C D E F G H I J K L P Figure 8.76
RkJQdWJsaXNoZXIy NjM5ODQ=