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Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce

Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce. Geometric Transformations:. Pre-Image: original figure Image : after transformation. Use prime notation. Notation:. A’. C. C ’. B. B’. A. Isometry. AKA: congruence transformation

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Translation: slide Reflection: mirror Rotation: turn Dialation: enlarge or reduce

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  1. Translation:slideReflection: mirrorRotation:turnDialation:enlarge or reduce Geometric Transformations:

  2. Pre-Image: original figureImage: after transformation. Use primenotation Notation: A’ C C ’ B B’ A

  3. Isometry AKA: congruence transformation a transformation in which an original figure and its image are congruent.

  4. Theorems about isometries FUNDAMENTAL THEOREM OF ISOMETRIES Any any two congruent figures in a plane can be mapped onto one another by at most 3 reflections ISOMETRY CLASSIFICATION THEOREM There are only 4 isometries. They are:

  5. TRANSLATION:moves all points in a planea given directiona fixed distance

  6. TRANSLATION VECTOR:DirectionMagnitude PRE-IMAGE IMAGE

  7. Translate by the vector <x, y>

  8. x moves horizontaly moves vertical Translate by <3, 4>

  9. Different notationT(x, y) -> (x+3, y+4)

  10. Translations PRESERVE:SizeShapeOrientation

  11. Reflectionover a line (mirror)

  12. Properties of reflections PRESERVE • Size (area, length, perimeter…) • Shape CHANGE orientation (flipped)

  13. Reflect x-axis: (a, b) -> (a,-b)Change sign y-coordinate

  14. Reflect y-axis: (a, b) -> (-a, b)Change sign on x coordinate

  15. X-axis reflection

  16. Y-axis reflection

  17. PARTNER SWAP:Part I: (Live under my rules) • Use sketchpad to graph & label any three points • Graph & Reflect them over the line y = x • Graph->Plot new function->x->OK • Construct two points on the line and connect them • Mark this line segment as your mirror. • WRITE a conjecture about how (a, b) will be changed after reflecting over y = x. Explain. • Repeat by reflecting over the line y = -x. Write a conjecture.

  18. Starter: • Find one vector which would accomplish the same thing as translating (3, -1) by <3, 8> then applying the transformation T(x, y)->(x-4, y+9) • Find coordinates of (7, 6) reflected over: a.) the y-axis b.) the x-axis c.) the line y = x d.) the line x = -3 3. HW Check & Peer edit

  19. Rotations have: Center of rotation Angle of rotation:

  20. Example: Rotate Triangle ABC 60 degrees clockwise about “its center” • Find the image of A after a 120 degree rotation • Find the image of A after a 180 degree rotation • Find the image of A after a 240 degree rotation • Find the image of A after a 300 degree rotation • Find the image of A after a 360 degree rotation

  21. Rotated 90 degrees counterclockwise

  22. ROTATIONS PRESERVE SIZE • Length of sides • Measure of angles • Area • Perimeter SHAPE ORIENTATION

  23. PARTNER SWAP:Part II: (Live under new rules) • Use sketchpad to graph & label any three points. Connect them and construct triangle interior. • Rotate your pre-image about the origin 90 • Rotate the pre-image about the origin 180 • Rotate the pre-image about the origin 270 • Rotate the pre-image about the origin 360 WRITE A CONJECTURE: What are the coordinates of (a, b) after a 90, 180, and 270 degree rotation about the origin?

  24. Rotations on a coordinate plane about the origin 90 (a, b) -> (-b, a) 180 (a, b) -> (-a, -b) 270 (a, b) -> (b, -a) 360 (a, b) -> (a, b)

  25. DEBRIEFING: Find the coordinates of (2, 5) • Reflected over the x-axis • Reflected over the y-axis • Reflected over the line x = 3 • Reflected over the line y = -2 • Reflected over the line y = x • Rotated about the origin 180 • Rotated about the origin 270  • Rotated about the origin 360 

  26. Review the rules for coordinate geometry transformations • Which two transformations would accomplish the same thing as a 90 degree rotation about the origin? • Use sketchpad to justify your answer

  27. Coordinate Geometry rules Reflections x axis (a, b) -> (a, -b) y axis (a, b) -> (-a, b) y=x (a, b) -> (b, a) Rotations about the origin 90 (a, b) -> (-b, a) 180 (a, b) -> (-a, -b) 270 (a, b) -> (b, -a) 360 (a, b) -> (a, b)

  28. GLIDE REFLECTIONSYou can combine different Geometric Transformations…

  29. Practice: Reflect over y = x then translate by the vector <2, -3>

  30. After Reflection…

  31. After Reflection and translation…

  32. Santucci’s Starter: Complete the following transformations on (6, 1) and list coordinates of the image: a. Reflect over the x-axis b. Reflect over the y-axis c. Rotate 90 about the origin d. Rotate 180 about the origin e. Rotate 270 about the origin EXPLAIN in writing: what two transformations would accomplish the same thing as a 90 degree rotation about the origin?

  33. Starter: Find the coordinates of pre-image (3, 4) after the following transformations (do without graphing…) • reflect over y-axis • reflect over x-axis • reflect over y=x • reflect over y=-x • translate <-2, 6> • rotate 90 about origin • rotate 180 about origin • rotate 270 about origin • rotate 360 about origin

  34. PAIRS Sketchpad Exploration: • Rotate (3, 4) 90 degrees about the point (1, 6). What two transformations will produce the same result? • Try it again by rotating (3, 4) 90 degrees about (-2, 5). • Rotate (2, -6) 90 degrees about (1, 7) • Describe OR LIST STEPS FOR how you can find the image of any point after a 90 rotation about (a, b). • Try it again with a 180 rotation about (a,b). How can you find the image? • Try it again with a 270 rotation about (a,b). How can you find the image?

  35. Starter HW Peer edit Practice 12-5 • Reflectional symmetry • Reflectional symmetry • Both rotational and Reflectional symmetry • Reflectional symmetry • See key • See key • No lines of symmetry • Line symmetry (5 lines) and 72 degree rotational symmetry • Line symmetry (1 line) • Line symmetry (4 lines) and 90 degree rotational symmetry • Line symmetry (8 lines) and 45 degree rotational symmetry • 180 degree rotational symmetry • Line symmetry (1 line) • Line symmetry (8 lines) and 45 degree rotational symmetry • 180 degree rotational symmetry • Line symmetry (1 line) #17-21 see key

  36. Symmetry Line Symmetry If a figure can be reflected onto itself over a line. Rotational Symmetry If a figure can be rotated about some point onto itself through a rotation between 0 and 360 degrees

  37. What kinds of symmetry do each of the following have?

  38. What kinds of symmetry do each of the following have? Rotational (180) Point Symmetry Rotational (90, 180, 270) Point Symmetry Rotational (60, 120, 180, 240, 300) Point Symmetry

  39. Isometry Wrap Up… • Sketchpad Activitiy # 6 Symmetry in Regular Polygons • Dilations Exploration • NOTE: TEST WILL BE END OF NEXT WEEK!!!

  40. Dilations • Plot any 5 points to make a convex polygon and fill in its interior red. • Mark the origin as center. • Make the polygon larger by a scale factor of 2 and fill it in green. • Make the polygon smaller by a scale factor of 1/3. Fill it in red. • Measure your coordinates and Explain how you can find coordinates of a dilation image. • Try marking a new center and dilating a few points. What is the “center” of a dilation? How does it change the measurements?

  41. Tessellations web-quest VISIT: http://www.tessellations.org/tess-what.htm Explore & read information underTessellations: What are they The beginnings Symmetry & MC Escher The galleries Solid Stuff Answer the following questions: 1. What is symmetry and list the types discussed. 2. What are the Polya’ symmetries? 3. How many Polya’ symmetries are there? 4. What are the Rhomboid possibilities? 5. What is the difference between a periodic and aperiodic tiling?

  42. TO-DO • Complete Tessellations Sketchpad explorations, # 8, 9 • Read rubric and write questions. Begin design

  43. INDIRECT PROOFIf ~q then ~p • Assume that the conclusion isFALSE. • Reason to a contradiction. If n>6 then the regular polygon will not tessellate. ASSUME:The polygon tessellates SHOW:n can not be >6

  44. Indirect proofRegular polygons with n>6 sides will not tessellate Proof: • Assume a polygon with n>6 sides will tessellate. • This means that n*one interior <measure will equal 360 • IF n = 3 there are 6 angles about center point • IF n = 4 there are 4 angles about center point • IF n = 6 there are 3 angles about center point • Therefore, if n>6 then there must be fewer than 3 angles about the center point. In other words, there must be 2 or fewer. If there are 2 angles about the center point then each angle must measure 180 to sum to 360 • But no regular polygon exists whose interior angle measures 180 (int. < sum must be LESS than 180). Therefore, the polygon can not tessellate.

  45. Santucci’s Starter Determine if the following will tessellate & provide proof: • Isosceles triangle • Kite • Regular pentagon • Regular hexagon • Regular heptagon • Regular octagon • Regular nonagon • Regular decagon

  46. Review practice • Find the image of A(-1, 4) reflected over the x-axis then over the y-axis (two intersecting lines). What one transformation would accomplish the same result? • Find the image of B(6, -2) reflected over x=3 then over x=-5 (two parallel lines). What one transformation would accomplish the same result? • List all the rotational symmetries of a regular decagon. • Draw a regular octagon with all its lines of symmetry (on sketchpad).

  47. Problem from HSPA test

  48. Coordinate TransformationsMOAT game Groups of “3” Write answer on white board and send one “runner” to stand facing the class with representatives from all other groups (hold board face down). When MOAT is called flip answer so all members seated can see answer. 1st group correct = +3 points 2nd group correct = +2 points 3rd group correct = +1 points Group with HIGHEST # points +3 on quiz Group with 2nd highest # points +2 on quiz Group with 3rd highest # points +1 on quiz

  49. HW Answers p. 650 10. H 11. M 12. C 13. Segment BC 14. A 15. Segment LM 16. I 17. K 34. a.) B(-2, 5) b.) C(-5, -2) c.) D(2, -5) d.) Square: 4 congruent sides & angles • 12-4 • 4. F translate twice the distance • Translate T across m twice the distance between l and m • V rotated 145 • 10-17. Peer edit • opp; reflection • same; translation • same; 270 rotation • opp; reflection • Glide <-2, -2>, reflect over y = x – 1 • 28. Glide <0, 4>, reflect over y = 0 (x-axis)

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