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Transformations 3-6, 3-7, & 3-8

Understand slides, flips, and turns in a plane through translations, reflections, and rotations. Learn how to write rules for each transformation and identify new coordinates of points after moving. Explore symmetry in figures and rotations about fixed points in a coordinate plane.

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Transformations 3-6, 3-7, & 3-8

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  1. Transformations3-6, 3-7, & 3-8

  2. Transformation • Movements of a figure in a plane • May be a SLIDE, FLIP, or TURN • a change in the position, shape, or size of a figure.

  3. Image The figure you get after a translation A A’ Slide C B C’ B’ Image Original To identify the image of point A, use prime notation Al. You read Al as “A prime”. The symbol ‘ is read “prime”. ABC has been moved to A’B’C’. A’B’C’ is the image of ABC.

  4. Translation • a transformation that moves each point of a figure the same distance and in the same direction. AKA - SLIDE A A’ C C’ B B’

  5. Writing a Rule for a Translation Finding the amount of movement LEFT and RIGHT and UP and DOWN

  6. Writing a Rule Right 4 (positive change in x) B Down 3 (negative change in y) B’ C A A’ C’

  7. Writing a Rule Can be written as: R4, D3 (Right 4, Down 3) Rule: (x,y) (x+4, y-3)

  8. Translations Example 1: If triangle ABC below is translated 6 units to the right and 3 units down, what are the coordinates of point Al. A (-5, 1) B (-1, 4) C (-2, 2) Rule (x+6, y-3) -First write the rule and then translate each point. Cl = (4, -1) Al = (1, -2) Bl = (5, 1) -Now graph both triangles and see if your image points are correct. B A C B’ C’ A’

  9. Example 2: Triangle JKL has vertices J (0, 2), K (3, 4), L (5, 1). Translate the triangle 4 units to the left and 5 units up. What are the new coordinates of Jl? -First graph the triangle and then translate each point. Kl = (-1, 9) Ll = (1, 6) Jl = (-4, 7) -You can use arrow notation to describe a translation. For example: (x, y) (x – 4, y + 5) shows the ordered pair (x, y) and describes a translation to the left 4 unit and up 5 units. K’ J’ L’ K J L

  10. You try some: Graph each point and its image after the given translation. a.) A (1, 3) left 2 units b.) B (-4, 4) down 6 units Al (-1, 3) Bl (-4, -2) B Al A Bl

  11. Example 3: Write a rule that describes the translation below Point A (2, -1) Al (-2, 2) Point B (4, -1) Bl (0, 2) Point C (4, -4) Cl (0, -1) Point D (2, -4) Dl (-2, -1) Rule (x, y) (x – 4, y + 3) Example 4: Write a rule that describes each translation below. a.) 3 units left and 5 units up b.) 2 units right and 1 unit down Rule (x, y) (x – 3, y + 5) Rule (x, y) (x + 2, y – 1)

  12. Reflection Another name for a FLIP A A’ C B B’ C’

  13. Reflection Used to create SYMMETRY on the coordinate plane

  14. Symmetry When one side of a figure is a MIRROR IMAGE of the other

  15. Line of Reflection The line you reflect a figure across Ex: X or Y axis X - axis

  16. In the diagram to the left you will notice that triangle ABC is reflected over the y-axis and all of the points are the same distance away from the y-axis. Therefore triangle AlBlCl is a reflection of triangle ABC Example 1: Draw all lines of reflection for the figures below. This is a line where if you were to fold the two figures over it they would line up. How many does each figure have? a.) b.) 1 6

  17. Example 2: Graph the reflection of each point below over each line of reflection. a.) A (3, 2) is reflected over the x-axis b.) B (-2, 1) is reflected over the y-axis A B Bl Al

  18. Example 3: Graph the triangle with vertices A(4, 3), B (3, 1), and C (1, 2). Reflect it over the x-axis. Name the new coordinates. C A B C’ (1,-2) B’ (3,-1) A’ (4,-3)

  19. Symmetry of the Alphabet • Sort the letters of the alphabet into groups according to their symmetries • Divide letters into two categories: • symmetrical • not symmetrical

  20. Symmetry of the Alphabet • Symmetrical: A, B, C, D, E, H, I, K, M, N, O, S, T, U, V, W, X, Y, Z • Not Symmetrical: F, G, J, L, P, Q, R

  21. Rotation Another name for a TURN C’ B’ B A’ C A

  22. Rotation A transformation that turns about a fixed point

  23. Center of Rotation The fixed point C’ B’ B A’ C A (0,0)

  24. Rotating a Figure Measuring the degrees of rotation C’ B’ B 90 degrees A’ C A

  25. Rotations in a Coordinate Plane This transformation can be described as (x, y) (–y, x). In a coordinate plane, sketch the quadrilateral whose vertices are A(2, –2), B(4, 1), C(5, 1), and D(5, –1). Then, rotate ABCD 90º counterclockwise about the origin and name the coordinates of the new vertices. Describe any patterns you see in the coordinates. SOLUTION Plot the points, as shown in blue. Use a protractor, a compass, and a straightedge to find the rotated vertices. The coordinates of the preimage and image are listed below. In the list, the x-coordinate of the image is the opposite of the y-coordinate of the preimage. The y-coordinate of the image is the x-coordinate of the preimage. Figure A'B'C'D' Figure ABCD A(2, –2) A'(2, 2) B(4, 1) B'(–1, 4) C(5, 1) C'(–1, 5) D(5, –1) D'(1, 5)

  26. Rotational symmetry can be found in many objects that rotate about a centerpoint. Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

  27. Hubcap 1 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

  28. Hubcap 1 There are 5 lines of symmetry in this design. 360 degrees divided by 5 =

  29. Hubcap 1 72º The angle of rotation is 72º.

  30. Hubcap 2 There are NO lines of symmetry in this design.

  31. Hubcap 2 120º The angle of rotation is 120º. (360 / 3) There are NO lines of symmetry in this design.

  32. Hubcap 3 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

  33. Hubcap 3 There are 10 lines of symmetry in this design. 360 / 10 = 36 However to make it look exactly the same you need to rotate it 2 angles. 36 x 2 = 72

  34. Hubcap 3 36º The angle of rotation is 36º. There are 10 lines of symmetry in this design.

  35. Hubcap 4 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

  36. Hubcap 4 . There are 9 lines of symmetry in this design.

  37. Hubcap 4 40º The angle of rotation is 40º. There are 9 lines of symmetry in this design.

  38. Think About it: Is there a way to determine the angle of rotation for a particular design without actually measuring it?

  39. When there are lines of symmetry 360 ÷ number of lines of symmetry = angle of rotationWhen there are no lines of symmetry: 360 ÷ number of possible rotations around the circle. 5 lines of symmetry 3 points to rotate it to

  40. Homework • Pg 138 #8, 12, 18, & 22 • Pg 143 #8, 10, 16, & 18 • Pg 148 #6, 8, 10

  41. Tessellation A design that covers a plane with NO GAPS and NO OVERLAPS

  42. Tessellation Formed by a combination of TRANSLATIONS, REFLECTIONS, and ROTATIONS

  43. Pure Tessellation A tessellation that uses only ONE shape

  44. Pure Tessellation

  45. Pure Tessellation

  46. Semiregular Tessellation A design that covers a plane using more than one shape

  47. Semiregular Tessellation

  48. Semiregular Tessellation

  49. Semiregular Tessellation

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