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18.4. Name the translation that moves everything (a) Down 5 spaces (b) To the right 6 spaces
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CHAPTER 18 Transformations
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(c) Up 3 spaces and 7 spaces to the left (d) Down 2 spaces and 8 spaces to the right (e) Up 4 spaces and to the left 1 space
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18.5. Let trapezoid ABCD be formed by A(1, 3), B(5, 3), C(4, 1), and D(2, 1). Graph trapezoid ABCD and its image under (a) reflection across the y axis P(x, y) A P ( x, y) , (b) reflection across the line y 1, P(x, y) A Ps(x, 2 y), and (c) reflection across the line x 8, P(x, y) A P-(16 x, y). 18.6. Name the reflection that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs, and (c) ^A-B-C- as illustrated in Fig. 18-18.
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Fig. 18-18
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18.7. Name the transformation that (a) Reflects across y (b) Reflects across x (c) Reflects across y (d) Reflects across x 5 2 1
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18.8. Which of the figures in Fig. 18-19 has reflectional symmetry
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Fig. 18-19
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18.9. Let parallelogram ABCD be defined by A(1, 2), B(4, 2), C(5, 1), and D(2, 1). Graph parallelogram ABCD and its image under (a) a 90 clockwise rotation about the origin, (b) a 180 rotation about the origin, and (c) a 270 clockwise rotation about the origin.
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CHAPTER 18 Transformations
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Name the rotation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-20.
Fig. 18-20
18.11. Name the transformation that rotates clockwise about the origin: (a) 40 (b) 50 (c) 80 18.12. For each figure in Fig. 18-21, give the smallest angle by which the figure could be rotated around its center and still look the same.
Fig. 18-21
18.13. Let triangle ABC be defined by A(2, 1), B(3, 2), and C(3, 1). Graph ^ABC and its image under the following combinations of transformations: (a) Reflect across the line y 3 and then move to the right 2 spaces.
(b) Rotate about the origin 90 clockwise and then move to the right 1 space and down 3 spaces.
CHAPTER 18 Transformations
(c) Rotate about the origin 270 clockwise and then reflect across the x axis. (d) Reflect across the line x 1 and then move up 2 spaces.
18.14. Name the single transformation that does the same thing as (a) P(x, y) A Pr(x (b) P(x, y) A Pr(5 5, y 3) and then Qr(x, y) A Qs(x 4, y 3, y y) x, y) 6) 1, y 2) 2)
x, y) and then Qr(x, y) A Qs(x
(c) P(x, y) A Pr(y, x) and then Qr(x, y) A Qs(x (d) P(x, y) A Pr( y, x) and then Qr(x, y) A Qs(x, 4 (e) P(x, y) A Pr(x, 3
y) and then Qr(x, y) A Qs(6
18.15. Name the transformation that takes ^ABC to (a) ^ArBrCr, (b) ^AsBsCs , and (c) ^A-B-C- as illustrated in Fig. 18-22.
Fig. 18-22
18.16. Name the transformation that (a) Reflects across the x axis and then moves everything down 3 spaces (b) Rotates around the origin clockwise 90 and then moves everything to the right 2 spaces (c) Reflects across the line y 2 and then rotates 180 around the origin 2 4
(d) Rotates 180 around the origin and then reflects across the line y
(e) Moves everything up 3 spaces and to the left 1 space, then reflects across the line x
18.17. Let rectangle ABCD be formed by A( 1, 2), B(1, 2), C(1, 1), and D( 1, 1). Graph this rectangle and also its image under the transformation P(x, y) A Pr(3x, 3y). 18.18. Name the transformation that (a) Scales everything to be twice as large (b) Scales everything by scale factor 8 (c) Dihilates everything by a scale factor of 3
Non-Euclidean Geometry
19.1 The Foundations of Geometry
For most of the years since Euclid wrote the Elements in 325 B.C., people felt that only one sort of geometry was possible. Planes looked like infinitely large, flat sheets of paper, lines went on forever as straight as the mind could imagine, and a grid of parallel lines could be drawn to make a plane look like graph paper. However, the foundations of this geometry were unfortunately vague. As discussed in 1, the concepts of point, line, and plane were not given formal definitions. The individual points, infinite straight lines, and flat planes discussed throughout this book all fit the properties and descriptions of points, lines, and planes, but other objects fit these general descriptions as well. When these basic objects are different, the resulting geometry is different as well. Similarly, as discussed in 2, the entire structure of geometric proof rests upon unproved postulates. These postulates lead to the geometry with which we are familiar. However, why should we believe one set of postulates and not a different set If we start with different postulates, then our theorems will be different as well. Our choices lead to different sorts of geometry, called non-euclidean geometries.