508 CHAPTER 8 Geometry Thus far in our examples of rotations, the rotation point was outside the figure being rotated. We now will study an example in which the rotation point is inside the figure to be rotated. A0 C0 B0 D0 P A B C D Figure 8.64 c) To gain some perspective on a 270° rotation, rotate this book or a printed page 270° in the counterclockwise direction. The rotated rectangle A B C D ′′′ ′′′ ′′′ ′′′ is shown along with rectangle ABCD in Fig. 8.65. 7 Now try Exercise 27 A B C D D- AB- CP Figure 8.65 A B P C D Figure 8.66 A D C P B B9 C9 A9 D9 Figure 8.67 C0 D0 P B0 A0 Figure 8.68 Example 5 A Rotation Point Inside a Parallelogram Given parallelogram ABCD and rotation point P, shown in Fig. 8.66, construct parallelograms that result from rotations through a) 90.° b) 180 .° Solution a) We will rotate the parallelogram 90° in a counterclockwise direction. The resulting parallelogram will be the same size and shape as parallelogram ABCD. To visualize what the rotated parallelogram will look like, pick up this book or a printed page and rotate it counterclockwise 90 .° Figure 8.67 shows the original parallelogram, ABCD, in pale blue, and the rotated parallelogram, A BC D, ′ ′ ′ ′ in darker blue. Notice how line segments AB and CD in parallelogram ABCD are horizontal, but in the rotated parallelogram A BC D, ′ ′ ′ ′ line segments A B′ ′ and C D′ ′ are vertical. Also notice in the original parallelogram ABCD that line segment AB is one unit above the rotation point P, but in the rotated parallelogram A BC D, ′ ′ ′ ′ line segment A B′ ′ is 1 unit to the left of rotation point P. b) To visualize the parallelogram obtained through a 180° rotation, pick up this book or a printed page and rotate it 180° in the counterclockwise direction. Notice from Fig. 8.68 that vertex A″ of the rotated parallelogram is in the same position as vertex C of the original parallelogram. In fact, each of the vertices of the rotated parallelogram A B C D ″ ″ ″ ″ is in the same position as a different vertex in the original parallelogram ABCD (see Fig. 8.68). 7 Now try Exercise 31
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