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Recognizing reflectional symmetry Which of the figures in Fig. 18-8 have reflectional symmetry
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Fig. 18-8
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CHAPTER 18 Transformations
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Only (b), (e), and (f) have reflectional symmetry. Note that when (d) is flipped, it will look like Fig. 18-9, which is different from the original in that the upper-left-hand crossing is horizontal instead of vertical.
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Fig. 18-9
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18.5 Rotations
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If a pin were pushed through the origin on a graph and the paper were to be turned, the result would be a rotation about the origin. A rotation is described by the number of degrees by which the paper is turned. The 90 clockwise rotation (or 270 counter-clockwise) about the origin is given by P(x, y) A P (y, x). The 180 rotation about the origin is given by P(x, y) A P (x, y). The 270 clockwise (or 90 counter-clockwise) rotation about the origin is given by P(x, y) A P ( y, x). In general, the clockwise rotation about the origin of is given by P(x, y) A P (x cos y sin , y cos x sin )
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Performing rotations Let triangle ABC be given by A(2, 1), B(3, 1), and C(3, 4). Graph the image of ^ABC as rotated about the origin by (a) 90 clockwise, (b) 180 , and (c) 270 clockwise.
Solutions
See Fig. 18-10.
Fig. 18-10
CHAPTER 18 Transformations
(a) Ar(1, 2), Br(1, 3), and Cr(4, 3) (b) As( 2, 1), Bs( 3, 1), Cs( 3, 4) (c) A-( 1, 2), B-( 1, 3), C-( 4, 3)
18.10 Recognizing rotations Name the rotation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-11.
Fig. 18-11
Solutions
(a) 270 clockwise or 90 counter-clockwise about the origin, P(x, y) A Pr( y, x) (b) 180 about the origin (either clockwise or counter-clockwise), P(x, y) A Ps( x, y) (c) 45 clockwise, P(x, y) A P-(x cos 45 P-a 22 x 2 22 22 y, y 2 2 22 xb 2 y sin 45, y cos 45 x sin 45 )
18.11 Naming rotations Name the transformation that rotates clockwise about the origin: (a) 20 (b) 30 (c) 60 (d) 75
Solutions
(a) P(x, y) A Pr(x cos 20 (b) P(x, y) A Pr(x cos 30 Pr(0.866x y sin 20 , y cos 20 y sin 30 , y cos 30 0.5x) x sin 20 ) x sin 30 ) Pr(0.9397x Pr a 23 x 2 0.3420y, 0.9397y 1 23 y, y 2 2 1 xb 2 0.3420x)
0.5y, 0.866y
CHAPTER 18 Transformations
23 1 y, y 2 2 23 xb 2
(c) P(x, y) A Pr(x cos 60 Pr(0.5x 0.866y, 0.5y
y sin 60 , y cos 60 0.866) y sin 75 , y cos 75
x sin 60 )
1 Pra x 2
(d) P(x, y) A Pr(x cos 75
x sin 75 )
Pr(0.2588x
0.9659y, 0.2588y
0.9659x)
18.5A Rotational Symmetry
A figure has rotational symmetry if it can be rotated around its center by fewer than 360 and look the same as it did originally. In Fig. 18-12, there is (a) a figure that looks the same under a 72 rotation, (b) a figure that looks the same under a 120 rotation, (c) a figure that looks the same under a 180 , and (d) a figure without rotational symmetry.
Fig. 18-12
SOLVED PROBLEMS
18.12 Recognizing rotational symmetry For each figure in Fig. 18-13, give the smallest angle by which the figure could be rotated around its center and still look the same.
Fig. 18-13
CHAPTER 18 Transformations
Solutions
(a) 90 (b) 120 (c) 360 (no rotational symmetry) (d) 180 (e) 360 (no rotational symmetry) (f) 90
18.6 Rigid Motions
Any combination of translations, reflections, and rotations is called a rigid motion because figures are moved without changing angles, lengths, or shapes. The image of a figure under a rigid motion will always be congruent to the original.
SOLVED PROBLEMS
18.13 Graphing rigid motions Let triangle ABC be formed by A( 4, 2), B( 4, 1), and C( 1, 1). Graph ^ABC and its image under the following combinations of transformations: (a) Reflect across the y axis and then move to the right 4 spaces. (b) Rotate 90 clockwise around the origin then move up 3 spaces. (c) Reflect across y 2 then move up 2 spaces and to the left 3 spaces.
(d) Reflect across the x axis and then reflect across the y axis. (e) Rotate 90 counter-clockwise around the origin and then reflect across x
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