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See Fig. 18-14.
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Fig. 18-14
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CHAPTER 18 Transformations
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18.14 Combining transformations Name the single transformation that does the same thing as the combination of: (a) P(x, y) A Pr(x 7, y 2) and then Qr(x, y) A Qs(x 3, y x, y) 5, y 1) 3, y 2) 5) (b) P(x, y) A Pr( x, y) and then Qr(x, y) A Qs(x (c) P(x, y) A Pr( x, y) and then Qr(x, y) A Qs(4 (d) P(x, y) A Pr(y, x) and then Qr(x, y) A Qs(x
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(a) R(x, y) A Rs((x (b) R(x, y) A Rs( x (c) R(x, y) A Rs(4 (d) R(x, y) A Rs(y ( 5, 7) 3, y x), x 3, (y 2) y) 1) Rs(4 x, y) 2) 5) Rs(x 10, y 3)
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18.15 Recognizing rigid motions Name the transformation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-15.
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(a) The triangle has been reflected across the y axis and then moved up 1 space, so P(x, y)APr( x, y (b) The triangle has been reflected across the line y P(x, y) A Pr(x 2, 7 y). 1).
3, then moved up 1 space and to the left 2 spaces, so
(c) The triangle has been rotated clockwise around the origin 90 and then moved up 8 spaces and to the right 6 spaces, so P(x, y) A Pr(y 6, x 8).
18.16 Naming rigid motions Name the transformation that (a) Rotates everything around the origin 180 , then moves everything up 3 spaces (b) Reflects across x 4, then slides everything down 2 spaces
(c) Rotates everything 90 clockwise around the origin, then reflects across the y axis (d) Rotates around the origin 90 counter-clockwise, then slides to the left 5 spaces
CHAPTER 18 Transformations
Solutions
(a) P(x, y) A Pr( x, (b) P(x, y) A Pr(8 (d) P(x, y) A Pr( y y x, y 3) 2)
(c) P(x, y) A Pr( y, x) 5, x)
18.7 Dihilations
A dihilation (also called a scaling or an enlargement) is not a rigid motion because it multiplies all lengths by a single scale factor. The image of a figure under a dihilation will always be similar to the original. The dihilation that enlarges everything by a scale factor of k is P(x, y) A Pr(kx, ky).
SOLVED PROBLEMS
18.17 Performing dihilations Let triangle ABC be formed by A(2, 2), B(4, 2), and C(4, 3). Graph the image of ^ABC under 1 (a) magnification by 2 and (b) scaling by 2.
Solutions
See Fig. 18-16.
Fig. 18-16
(a) Ar(4, 4), Br(8, 4), and Cr(8, 6) (b) As(1, 1), Bs(2, 1), and Cs(2, 1.5)
18.18 Naming dihilations Name the transformation that (a) Scales everything 5 times larger (b) Shrinks every length to half size (c) Triples all linear dimensions (d) Depicts everything at 10 scale (e) Dihilates by a scale factor of 12
CHAPTER 18 Transformations
Solutions
(a) P(x, y) A Pr(5x, 5y)
1 1 (b) P(x, y) A Pr(2x, 2y)
(c) P(x, y) A Pr(3x, 3y)
1 1 (d) P(x, y) A Pr(10x, 10y)
(e) P(x, y) A Pr(12x, 12y)
SUPPLEMENTARY PROBLEMS
18.1. Name the image of points A(6, 2), B( 1, 4), and C(2, 7) under the transformation (a) P(x, y) A Pr(x (b) P(x, y) A Pr(2 (c) P(x, y) A Pr( x (d) P(x, y) A Pr( y, x (e) P(x, y) A Pr(2x, 2y) (f) P(x, y) A Pr(4 (g) P(x, y) A Pr(2 3x, 3y) y, 5 x) 1), B(2, 2), and C(3, 1). Graph the image of ^ABC under 3, y) x, y) 1, 8) y 3)
18.2. Let triangle ABC be defined by A(1, (a) P(x, y) A Pr(x (b) P(x, y) A Pr(x, y (c) P(x, y) A Pr(x 18.3. 5, y) 4) 3, y 2)
Name the translation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs, and (c) ^A-B-C- as illustrated in Fig. 18-17.
Fig. 18-17
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