2. Geometry Dilations

3. September 8, 2015 Goals Identify Dilations Make drawings using dilations.

4. September 8, 2015 Rigid Transformations Previously studied in Chapter 7. Rotations Translations These were isometries: The pre-image and the image were congruent.

5. September 8, 2015 Dilation Dilations are non-rigid transformations. The pre-image and image are similar, but not congruent.

6. September 8, 2015

7. Dilation E n l a r ge m e n t

8. September 8, 2015 Dilation R e d u c t io n

9. September 8, 2015 Dilation Center of Dilation R S T C

10. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R

11. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R CS S 2CS

12. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R CS S 2CS CT 2CT T

13. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R CS S 2CS CT 2CT T  RST ~  RST

14. September 8, 2015 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 1.

15. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R CS S 2CS CT 2CT T Enlargement Scale Factor

16. September 8, 2015 Dilation Center of Dilation R S T C CR 2CR R CS S 2CS CT 2CT T  RST ~  R’S’T’ Scale Factor:

17. September 8, 2015 Example F G H K F’ G’ H’ K’ C What type of dilation is this? Reduction

18. September 8, 2015 Example F G H K F’ G’ H’ K’ C What is the scale factor? 36 12 45 15 Notice: k < 1 Reduction

19. September 8, 2015 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

20. September 8, 2015 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.

21. September 8, 2015 Graph  ABC with A(1, 1), B(3, 6), C(5, 4). A B C Notice the origin is here

22. September 8, 2015 Using a scale factor of k = 2, locate points A’, B’, and C’. P(x, y)  P’(kx, ky). A B C A(1, 1)  A’(2  1, 2  1) = A’(2, 2) A’ B(3, 6)  B’(2  3, 2  6) = B’(6, 12) B’ C(5, 4)  C’(2  5, 2  4) = C’(10, 8) C’

23. September 8, 2015 Draw  ABC. A B C A’ B’ C’

24. September 8, 2015 You’re done. A B C A’ B’ C’ Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point.

25. September 8, 2015 Do this problem. R(0, 0) Draw RSTV with R(0, 0) S(  6, 3) T(0, 12) V(6, 3) S(-6, 3) T(0, 12) V(6, 3)

26. September 8, 2015 Do this problem. R(0, 0) Draw R’S’T’V’ using a scale factor of k = 1/3. S(-6, 3) T(0, 12) V(6, 3) R’(0, 0) S’(-2, 1) T’(0, 4) V’(2, 1)

27. September 8, 2015 Do this problem. R(0, 0) R’S’T’V’ is a reduction. S(-6, 3) T(0, 12) V(6, 3) R’(0, 0) S’(-2, 1) T’(0, 4) V’(2, 1)

28. September 8, 2015 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.

29. September 8, 2015 One more time… Scale Factor = Image Size Pre-image Size Scale Factor = After Before

30. September 8, 2015 Enlargement or Reduction? CP = 10 and CP’ = 20 Enlargement What is the Scale Factor? 2 k = CP’/CP = 20/10 = 2

31. September 8, 2015 Enlargement or Reduction? CP = 150 and CP’ = 15 Reduction What is the Scale Factor? 1/10 k = CP’/CP = 15/150 = 1/10

32. September 8, 2015 Enlargement or Reduction? CP = 20 and CP’ = 18 Reduction What is the Scale Factor? 9/10 k = CP’/CP = 18/20 = 9/10

33. September 8, 2015 Enlargement or Reduction? CP = 15 and CP’ = 18 Enlargement What is the Scale Factor? 6/5 k = CP’/CP = 18/15 = 6/5