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Geometry

Geometry . Dilations. Goals. Identify Dilations Make drawings using dilations. Rigid Transformations. Previously studied in Chapter 7. Rotations Translations These were isometries : The pre-image and the image were congruent. Dilation. Dilations are non-rigid transformations.

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Geometry

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  1. Geometry Dilations

  2. Goals • Identify Dilations • Make drawings using dilations.

  3. Rigid Transformations • Previously studied in Chapter 7. • Rotations • Translations • These were isometries: • The pre-image and the image were congruent.

  4. Dilation • Dilations are non-rigid transformations. • The pre-image and image are similar, but not congruent.

  5. Dilation

  6. Dilation Enlargement

  7. Dilation Reduction

  8. Dilation R S C T Center of Dilation

  9. Dilation R 2CR CR R S CR C T Center of Dilation

  10. Dilation R 2CR CR 2CS R S CS S CR CS C T Center of Dilation

  11. Dilation R 2CR CR 2CS R S CS S CR CS C CT T Center of Dilation CT T 2CT

  12. Dilation RST ~ RST R 2CR CR 2CS R S CS S CR CS C CT T Center of Dilation CT T 2CT

  13. Dilation Definition A dilation with center C and scale factor k is a transformation that maps every point P to a point P’ so that the following properties are true: 1. If P is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive number such that k  1 and 2. If P is the center point C, then P = P’. 3. The dilation is a reduction if 0 < k < 1, and an enlargement if k > 1.

  14. Dilation Enlargement R 2CR CR 2CS R S CS S CR CS C CT T Center of Dilation CT T 2CT Scale Factor

  15. Dilation RST ~ R’S’T’ R 2CR CR 2CS R S CS S CR CS C CT T Center of Dilation CT T 2CT Scale Factor:

  16. G F F’ G’ C H’ K’ H K Example Reduction What type of dilation is this?

  17. G F F’ G’ C H’ K’ H K Example What is the scale factor? 45 Notice: k < 1 Reduction 15 12 36

  18. Remember: • The scale factor k is • If 0 < k < 1 it’s a reduction. • If k > 1 it’s an enlargement. image segment pre-image segment

  19. Coordinate Geometry • Use the origin (0, 0) as the center of dilation. • The image of P(x, y) is P’(kx, ky). • Notation: P(x, y)  P’(kx, ky). • Read: “P maps to P prime” • You need graph paper, a ruler, pencil.

  20. Graph ABC with A(1, 1), B(3, 6), C(5, 4). B C Notice the origin is here A

  21. Using a scale factor of k = 2, locate points A’, B’, and C’. P(x, y)  P’(kx, ky). B’ A(1, 1)  A’(2  1, 2  1) = A’(2, 2) B(3, 6)  B’(2  3, 2  6) = B’(6, 12) C’ C(5, 4)  C’(2  5, 2  4) = C’(10, 8) B C A’ A

  22. Draw ABC. B’ C’ B C A’ A

  23. You’re done. B’ Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point. C’ B C A’ A

  24. Do this problem. T(0, 12) Draw RSTV with R(0, 0) S(6, 3) T(0, 12) V(6, 3) S(-6, 3) V(6, 3) R(0, 0)

  25. Do this problem. T(0, 12) Draw R’S’T’V’ using a scale factor of k = 1/3. T’(0, 4) S(-6, 3) V(6, 3) S’(-2, 1) V’(2, 1) R(0, 0) R’(0, 0)

  26. Do this problem. T(0, 12) R’S’T’V’ is a reduction. T’(0, 4) S(-6, 3) V(6, 3) S’(-2, 1) V’(2, 1) R(0, 0) R’(0, 0)

  27. Summary • A dilation creates similar figures. • A dilation can be a reduction or an enlargement. • If the scale factor is less than one, it’s a reduction, and if the scale factor is greater than one it’s an enlargement.

  28. One more time… After Scale Factor = Before Image Size Scale Factor = Pre-image Size

  29. Enlargement or Reduction? • CP = 10 and CP’ = 20 • Enlargement • What is the Scale Factor? • 2 • k = CP’/CP = 20/10 = 2

  30. Enlargement or Reduction? • CP = 150 and CP’ = 15 • Reduction • What is the Scale Factor? • 1/10 • k = CP’/CP = 15/150 = 1/10

  31. Enlargement or Reduction? • CP = 20 and CP’ = 18 • Reduction • What is the Scale Factor? • 9/10 • k = CP’/CP = 18/20 = 9/10

  32. Enlargement or Reduction? • CP = 15 and CP’ = 18 • Enlargement • What is the Scale Factor? • 6/5 • k = CP’/CP = 18/15 = 6/5

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