Geometry slides Year 9 NZ


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Covers angles to transformations

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Geometry slides Year 9 NZ

  1. 1. Angles within a right angle add to 90° <ul><li>They are called complementary angles </li></ul>60° a=? a= 30°
  2. 2. Isosceles Triangles <ul><li>The angles opposite the equal sides are equal angles. </li></ul>73° b= ? b= 73°
  3. 3. Polygons Closed figures made up of straight sides.
  4. 4. Examples of Polygons <ul><li>Triangle </li></ul><ul><li>Quadrilateral </li></ul><ul><li>Pentagon </li></ul><ul><li>Hexagon </li></ul><ul><li>Octagon </li></ul><ul><li>Decagon </li></ul>
  5. 5. Angles of Polygons <ul><li>Interior angles- are angles </li></ul><ul><li>between the sides of the </li></ul><ul><li>polygon on the inside. </li></ul><ul><li>Exterior angles- are angles found by extending the sides of the polygon. </li></ul>
  6. 6. Exterior Angles <ul><li>Measure the exterior angles of your polygons. </li></ul><ul><li>Add the exterior angles of each shape together. </li></ul><ul><li>What do they add to? </li></ul>The sum of the exterior angles of a polygon is 360°. d Shape Total Degrees of Exterior Angles Triangle Quadrilateral Pentagon Hexagon 360 360 360 360
  7. 7. Alternate angles When two parallel lines are cut by a third line, then angles in alternate positions equal in size. Co-interior angles When two parallel lines are cut by a third line, co-interior angles are supplementary. Angles at a point. The sum of the sizes of the angles at a point is 360 Adjacent angles on a straight line The sum of the sizes of the angles on a line is 180 degrease
  8. 8. Adjacent angles in a right angle The sum of the size's of the angles in around different points but the same angle Vertically opposite angles Vertically opposite angles are equal in size. Corresponding angles When two parallel lines are cut by a third line, then angles in corresponding positions are equal in size.
  9. 9. Interior Angles of a Polygon Degrees in a triangle sum to 180° If there are 180 degrees in a triangle how many degrees must there Be in a quadrilateral which is split into 2 triangles. The rule (n­2) × 180° n is the number of sides of the polygon ttrsgf No. of sides of polygon 3 4 5 You do 6, 8, 10 Drawing Number of triangles 1 2 3
  10. 10. Regular Polygons <ul><li>A polygon is called regular if all its sides are the same length and all its angles are the same size. </li></ul><ul><li>e.g. equilateral triangle, a square or a regular pentagon. </li></ul>
  11. 11. 3 4 5 6 8 10 3 4 5 6 8 10 360 360 360 360 360 360 120 90 72 60 45 36 180 360 540 720 1080 1440 60 90 108 120 135 144 Exterior Angles Number of sides Sum of exterior angles Each exterior angle Equilateral triangle Square Pentagon Hexagon Octagon Decagon Interior Angles Number of sides Sum of exterior angles Each exterior angle Equilateral triangle Square Pentagon Hexagon Octagon Decagon
  12. 12. Navigation and Bearings
  13. 13. Bearings <ul><li>Bearings are angles which are measured clockwise from north. They are always written using 3 digits. </li></ul><ul><li>The bearings start at 000 facing north and finish at 360 facing north. </li></ul>Bearings 045 120 180 270
  14. 14. Bearings
  15. 18. Starter
  16. 25. Paper Cutting
  17. 27. Translation <ul><li>A translation is a movement in which each point moves in the same direction by the same distance. </li></ul><ul><li>To translate an object all you need to know is the image of one point. Every other point moves in the same distance in the same direction. </li></ul>
  18. 28. E D A B F G A' C E’ F’ B’ D’ C’ G’
  19. 29. Reflection <ul><li>In a reflection, and object and its image are on opposite sides of a line of symmetry. </li></ul><ul><li>This line is often called a mirror line. </li></ul>m
  20. 30. m
  21. 31. Where would the mirror line go?? m
  22. 32. Invariant Points <ul><li>If a point is already on the mirror line, it stays where it is when reflected. These points are called invariant points. </li></ul>
  23. 33. m m
  24. 34. Exercises Today <ul><li>Ex 26.03 2 of 1a, b, c or d. Page397 </li></ul><ul><li>26.04 Question 2, 7 and 8. Page 401 & 402 </li></ul><ul><li>Any three questions from 26.05. Page 404 </li></ul>
  25. 35. Rotation <ul><li>Rotation is a transformation where an object is turned around a point to give its image. </li></ul><ul><li>Each part of the object is turned through the same angle. </li></ul><ul><li>To rotate an object you need to know where the center of rotation is and the angle of rotation. </li></ul>
  26. 36. The angle of rotation <ul><li>This can be given in degrees or as a fraction such as a quarter turn. </li></ul><ul><li>The direction the object is turned can be either clockwise or anti-clockwise. </li></ul>
  27. 37. D A C B D’ A’ B’ C’
  28. 38. Starter <ul><li>For the numbers below think of what they could mean in the world. Get as many answers as possible. </li></ul>366 4 000 000 30 15 25 11
  29. 39. Rotation
  30. 40. Rotation <ul><li>In rotation every point rotates through a certain angle. </li></ul><ul><li>The object is rotated about a fixed point called the centre of rotation . </li></ul><ul><li>Rotation is always done in an anti-clockwise direction. </li></ul><ul><li>A point and it’s image are always the same distance from the centre of rotation. </li></ul><ul><li>The centre of rotation is the only invariant point. </li></ul>
  31. 41. What are these equivalent angles of Rotation? <ul><li>270º Anti clockwise is _______ clockwise </li></ul><ul><li>180º Anti clockwise is ______ clockwise </li></ul><ul><li>340º Anti clockwise is _______ clockwise </li></ul>Rotations are always specified in the anti clockwise direction
  32. 42. Drawing Rotations A B C D B’ D’ C’ ¼ turn clockwise = 90º clockwise
  33. 43. Drawing Rotations A B C B’ ¼ Turn anti-clockwise = 90º Anti-clockwise C’
  34. 44. By what angle is this flag rotated about point C ? 180º Remember: Rotation is always measured in the anti clockwise direction! C
  35. 45. By what angle is this flag rotated about point C ? 90º C
  36. 46. By what angle is this flag rotated about point C ? 270º C
  37. 48. Questions <ul><li>Ex 26.07 Page 411 Qn 1, 2, 3, 7 and 8. </li></ul><ul><li>Ex 26.08 Page 412 Qn 1, 3, 4 and 6. </li></ul>
  38. 49. Answers 26.07 A D C B B’ C’ D’ A’ A A’ B C B’ C’ 1 2
  39. 50. A D B B’ C’ D’ A’ 3 C 7 D’ A D B’ C’ A’ B C
  40. 51. s s 8
  41. 52. Ex. 26.08 <ul><li>a) P b) R c) QS </li></ul><ul><li>180° </li></ul><ul><li>0° or 360° </li></ul><ul><li>8. a) R b) Q c) CB </li></ul>
  42. 53. Define these terms <ul><li>Mirror line </li></ul><ul><li>Centre of rotation </li></ul><ul><li>Invariant </li></ul><ul><li>What is invariant in </li></ul><ul><li>Reflection </li></ul><ul><li>rotation </li></ul>The line equidistant from an object and its image The point an object is rotated about Doesn’t change The mirror line Centre of rotation
  43. 54. <ul><li>Draw a rectangle. Have two units across and 3 going up. (Drawn below) </li></ul><ul><li>Label the rectangle A, B, C and D. </li></ul><ul><li>Reflect the object in the line CD. Label the image A’, B’, C’ and D’. </li></ul><ul><li>Rotate the image about point D’ 90° anticlockwise. Label this image A’’, B’’, C’’ and D’’. </li></ul><ul><li>Translate the 2 nd image by the vector . Label the rectangle A’’’’, B’’’’, C’’’’ and D’’’’. </li></ul>C A B D
  44. 55. m A C B D D’ A’ B’ C’ D’’ B’’ C’’ A’’ C’’’ B’’’ D’’’ A’’’
  45. 56. Rotational Symmetry <ul><li>A figure has rotational symmetry about a point if there is a rotation other than 360° when the figure can turn onto itself. </li></ul><ul><li>Order of rotational symmetry is the number of times a figure can map onto itself. </li></ul>
  46. 57. Order of rotational symmetry= Order of rotational symmetry= 3 4
  47. 58. Line Symmetry <ul><li>A shape has line symmetry if it reflects or folds onto itself. </li></ul><ul><li>The fold is called an axis of symmetry. </li></ul>Line symmetry= 2
  48. 59. Total order of symmetry <ul><li>The number of axes of symmetry plus the order of rotational symmetry. </li></ul>2+2=4
  49. 60. Mathematical Dance <ul><li>You must prepare a dance lasting 1 minute for tomorrow’s lesson. </li></ul><ul><li>You must use moves demonstrating reflection, rotation and translation in your dance. </li></ul><ul><li>You must have a sheet with your dance moves recorded using mathematical language. </li></ul><ul><li>One person in the group must call out the dance moves during the performance whilst dancing with their group. </li></ul>
  50. 61. Mathematical Dance <ul><li>You have 10 minutes today and 5 minutes tomorrow to prepare your masterpiece. </li></ul><ul><li>You can bring sensible music and consumes for your dance tomorrow. </li></ul>
  51. 62. Groups <ul><li>Group 1- Claudia, Martine, Amiee and Emily </li></ul><ul><li>Group2- Olivia, Bailey, Rachael and Elle </li></ul><ul><li>Group3- Emma H, Erin, Michal and Charlotte </li></ul><ul><li>Group4- Abby, Mia, Ashleigh and Brittany </li></ul><ul><li>Group5- Payton, Shannon, Tayla and Susan </li></ul>
  52. 63. 0 5 6 7 8 1 -8 -7 -6 -5 -4 -3 -2 -1 4 2 3 0 5 6 7 8 1 -8 -7 -6 -5 -4 -3 -2 -1 4 2 3 1. Move the red dot by the following values and state where it now lies. (-1), (4), (7), (-4) and (11). 1. Move the green dot by the following values and state where it now lies. (-6), (3), (-9), (-4) and (5).
  53. 64. Groups <ul><li>Group One- Kelly, Ruby, Bella, Chrisanna and Bianca </li></ul><ul><li>Group Two- Hannah B, Grace, Shanice, Grace and Georgia R </li></ul><ul><li>Group Three- Hannah C, Remy, Olivia, Kendyl and Sarah </li></ul><ul><li>Group Four- Kelsey, Shaquille, Kiriana, Cadyne and Claudia </li></ul><ul><li>Group Five- Lauran, Ashlee, Sophie, Georgia W and Emily S </li></ul><ul><li>Group Six- Esther, Emily M, Jemma, Amelia and Julia </li></ul>
  54. 65. Translations Each point moves the same distance in the same direction There are no invariant points in a translation (every point moves)
  55. 66. ← movement in the x direction (right and left) ← movement in the y direction (up and down) ( ) y x + + - - <ul><li>Vectors describe movement </li></ul>
  56. 67. Vectors <ul><li>Vectors describe movement </li></ul>Each vertex of shape EFGH moves along the vector ( ) -3 -6 To become the translated shape E’F’G’H’
  57. 68. <ul><li>Translate the shape ABCDEF by the vector to give the image A`B`C`D`E`F`. </li></ul>( ) - 4 - 2
  58. 69. Vectors -5 +4 +2 -6 -6 -2 +3 +4
  59. 73. Object Image 6 1 Object Image -1 2
  60. 74. Enlargement <ul><li>When an object is enlarged its size changes. </li></ul><ul><li>A scale factor tells us how much larger a shape is after it has been enlarged. </li></ul>
  61. 75. Scale factor of 2 Scale factor of 1/2
  62. 76. Scale Factor <ul><li>The scale factor tells us how much the lengths of an object are multiplied by to get the lengths of the image. </li></ul>