Example 1

1. Example 1 Use the coordinate mapping (x, y) → (x + 8, y + 3) to translate ΔSAM to create ΔS’A’M’.

2. Dilations Objectives: • To use dilations to create similar figures • To perform dilations in the coordinate plane using coordinate notation

3. Dilations A dilation is a type of transformation that enlarges or reduces a figure. The dilation is described by a scale factorand a center of dilation.

4. Dilations The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage.

5. Example 2 What happens to any point (x, y) under a dilation centered at the origin with a scale factor of k?

6. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

7. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor. Enlargement:k > 1.

8. Dilations in the Coordinate Plane You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor. Reduction:0 < k < 1.

9. Example 3 Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation. Is this a reduction or an enlargement?

10. Example 4 A graph shows PQR with vertices P(2, 4), Q(8, 6), and R(6, 2), and segment ST with endpoints S(5, 10) and T(15, 5). At what coordinate would vertex U be placed to create ΔSUT, a triangle similar to ΔPQR?

11. Example 5 Figure J’K’L’M’N’ is a dilation of figure JKLMN. Find the coordinates of J’ and M’.

13. Exploring Tessellations This Exploration of Tessellations will guide you through the following: Definition ofTessellation RegularTessellations Symmetry inTessellations TessellationsAround Us Semi-RegularTessellations View artistictessellationsbyM.C. Escher Create yourownTessellation

14. What is a Tessellation? A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.

15. Tessellations in the World Around Us: Brick Walls Floor Tiles Checkerboards Honeycombs Textile Patterns Art Can you think of some more?

16. Are you ready to learn more about Tessellations? Regular Tessellations Semi-RegularTessellations Symmetry inTessellations

17. Regular Tessellations Regular Tessellations consist of only one type of regular polygon. Do you remember what a regular polygon is? A regular polygon is a shape in which all of the sides and angles are equal. Some examples are shown here: Triangle Square Pentagon Hexagon Octagon

18. Regular Tessellations Which regular polygons will fit together without overlapping or leaving gaps to create a Regular Tessellation? Maybe you can guess which ones will tessellate just by looking at them. But, if you need some help, CLICK on each of the Regular Polygons below to determine which ones will tessellate and which ones won’t: Triangle Square Pentagon Hexagon Octagon

19. Regular Tessellations Does a Triangle Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES.

20. Regular Tessellations Does a Square Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES.

21. Regular Tessellations Does a Pentagon Tessellate? Gap The shapes DO NOT fit together because there is a gap. So the answer is NO.

22. Regular Tessellations Does a Hexagon Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES. Hexagon Tessellationin Nature

23. Regular Tessellations Does an Octagon Tessellate? Gaps The shapes DO NOT fit together because there are gaps. So the answer is NO.

24. Figures that Tessellate • Find the measure of an angle of a regular polygon using the following formula • If is a factor of 360, then the n-gon will tessellate

25. Regular Tessellations As it turns out, the only regular polygons that tessellate are: TRIANGLES SQUARES HEXAGONS Summary of Regular Tessellations: Regular Tessellations consist of only one type of regular polygon. The only three regular polygons that will tessellate are the triangle, square, and hexagon.

26. Are you ready to learn more about Tessellations? Regular Tessellations Semi-RegularTessellations Symmetry inTessellations

27. Semi-Regular Tessellations Semi-Regular Tessellations consist of more than one type of regular polygon. (Remember that a regular polygon is a shape in which all of the sides and angles are equal.) How will two or more regular polygons fit together without overlapping or leaving gaps to create a Semi-Regular Tessellation? CLICK on each of the combinations below to see examples of these semi-regular tessellations. Hexagon & Triangle Square & Triangle Hexagon, Square & Triangle Octagon & Square

28. Semi-Regular Tessellations Hexagon & Triangle Can you think of other ways to arrange these hexagons and triangles?

29. Semi-Regular Tessellations Octagon & Square Look familiar? Many floor tiles have these tessellating patterns.

30. Semi-Regular Tessellations Square & Triangle

31. Semi-Regular Tessellations Hexagon, Square, & Triangle

32. Semi-Regular Tessellations Summary of Semi-Regular Tessellations: Semi-Regular Tessellations consist of more than one type of regular polygon. You can arrange any combination of regular polygons to create a semi-regular tessellation, just as long as there are no overlaps and no gaps. What other semi-regular tessellations can you think of?

33. Symmetry in Tessellations The four types of Symmetry in Tessellations are: Rotation Translation Reflection Glide Reflection

34. Symmetry in Tessellations Rotation To rotate an object means to turn it around. Every rotation has a center and an angle. A tessellation possesses rotational symmetry if it can be rotated through some angle and remain unchanged. Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns. CLICK HERE to view someexamples of rotational symmetry. Back to Symmetry in Tessellations

35. Rotational Symmetry

36. Rotational Symmetry

37. Rotational Symmetry Back to Rotations

38. Symmetry in Tessellations Translation To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. A tessellation possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged. A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern. CLICK HERE to view someexamples of translational symmetry. Back to Symmetry in Tessellations

39. Translational Symmetry Back to Translations

40. Symmetry in Tessellations Reflection To reflect an object means to produce its mirror image. Every reflection has a mirror line. A tessellation possesses reflection symmetry if it can be mirrored about a line and remain unchanged. A reflection of an “R” is a backwards “R”. CLICK HERE to view someexamples of reflection symmetry. Back to Symmetry in Tessellations

41. Reflection Symmetry

42. Reflection Symmetry Back to Reflections

43. Symmetry in Tessellations Glide Reflection A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. A tessellation possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged. CLICK HERE to view someexamples of glide reflection symmetry. Back to Symmetry in Tessellations

44. Glide Reflection Symmetry

45. Glide Reflection Symmetry Back to Glide Reflections

46. Symmetry in Tessellations • Summary of Symmetry in Tessellations: • The four types of Symmetry in Tessellations are: • Rotation • Translation • Reflection • Glide Reflection • Each of these types of symmetry can be found in various tessellations in the world around us.

47. Exploring Tessellations We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.

48. Exploring Tessellations We have also learned about Regular Tessellations, Semi-Regular Tessellations, and the four types of Symmetry in Tessellations.

49. Create Your Own Tessellation! • Now that you’ve learned all about Tessellations, it’s time to create your own. • You can create your own Tessellation by hand, or by using the computer. It’s your choice!

50. M.C. Escher developed the tessellating shape as an art form *Escher was a graphic artist, who specialized in woodcuts and lithographs. * He was born MauritsCornelisEscher in 1898, in Leeuwarden, Holland. * His father wanted him to be an architect, but bad grades in school and a love of drawing and design led him to a career in the graphic arts.