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  1. 1. Transformations<br />BTEOTSSSBAT solve problems involving reflection, rotation, translation and enlargement<br />
  2. 2. Key words<br />Transformation<br />Translation<br />Image<br />Object<br />Reflection<br />Rotation<br />Origin<br />Enlargement<br />Congruent<br />Similar<br />
  3. 3. A transformation changes a shape.<br />There are four main types of transformations: <br /><ul><li>Reflections,
  4. 4. Rotations
  5. 5. Enlargements and
  6. 6. Translation</li></li></ul><li>Reflection<br />The triangles above are reflected in the dotted mirror line. <br />The image is the same distance from the mirror line as the object.<br />To describe a reflection we need a mirror line. <br />
  7. 7. Now try these 1<br />Draw the image of the objects in (a) and (b) on the sheet <br />– it may go outside of the grid. <br />2. Draw the triangle ABC such that A is the point (1, 1)<br /> B is the point (3, 1)<br /> C is the point (1, 2). <br /> (a) Reflect the triangle in the x-axis to obtain triangle A1B1C1.<br /> What are the co-ordinates of triangle A1B1C1?<br /> (b) Reflect the triangle in the y-axis to obtain triangle A2B2C2.<br /> What are the co-ordinates of triangle A2B2C2?<br />
  8. 8. Rotation<br />Rotations occur when a shape is rotated a specified angle around a centre of rotation.<br />
  9. 9. Example<br />Rotate this triangle 90° through the origin (0,0)<br />First mark the centre of rotation.<br />Draw around the original shape using tracing paper. <br />Rotate the tracing paper 90° clockwise around the centre of rotation, draw the new position of the image.<br />
  10. 10. Now try these 2<br />A triangle’s co-ordinates of the vertices are (2, 1), (1, 6), (2, 3).<br />Rotate it in the following ways (draw your answer on the grid above):<br />(a) 90 about (1, 0)<br />(b) 90 about (0, 1)<br />(c) 90 about (3, 0)<br />(d) 180 about (2, 0)<br />(e) 180 about (0, 0)<br />(f) 270 about (2, 1)<br />(g) 270 about (0, 2)<br />Note:all angles are anti-clockwise, this is how angles are given in rotations unless it says clockwise.<br />
  11. 11. Enlargement<br />The diagram shows two enlargements of an object A. <br /> The first is enlarged by a scale factor of 2, <br />the second by a scale factor of 4 from the centre of enlargement O.<br />The distance between O and A´ is 2  OA and <br />the distance between O and A´´ is 4  OA.<br />
  12. 12. Example <br />Enlarge the shape ABC with a scale factor of 3 from the centre of enlargement marked.<br />
  13. 13. Draw a line from the centre of enlargement going through each vertex of the shape.<br />
  14. 14. As the scale factor of enlargement is 3 then:<br />OA´ = 3 OA<br />OB´ = 3OB<br />OC´ = 3 OC<br />
  15. 15. Now try these <br />1. On the grid enlarge the shape by a scale factor of 3.<br />
  16. 16. 2. Enlarge the shape with a scale factor of 2 and centre (0,3)<br />
  17. 17. 3. T is an enlargement of S from a centre C.<br />On the grid mark the centre C and state the scale factor enlargement.<br />
  18. 18. Translation<br />The triangle above has been translated. It has moved 4 squares to the right and two squares up. The movement is shown by a vector:<br />movement in the x-direction<br />movement in the y-direction<br />In translation the size of the shape does not change, the shape is not rotated or reflected.<br />
  19. 19. Example<br />Describe the translation that moves the shaded shape to each of the other shapes.;<br />
  20. 20. Solution<br />To get to shape A it moves 6 to the right and 3 up <br />To get to shape B it moves 5 to the right and 5 down <br />To get to shape C it moves 5 to the left and 3 up <br />To get to shape D it moves 3 to the left and 4 down <br />
  21. 21. Now try these<br />1. Give the vector that translates the shaded shape to the other shapes.<br />
  22. 22.
  23. 23. Similarity and Congruence<br />Shapes are called congruent when they have the same shape and size. <br />If you translate, rotate or reflect a shape, the new shape will be congruent with the old one. <br />Shapes are called similar when they have the same shape but are different sizes. If you enlarge a shape, the new shape will be similar to the old one.<br />
  24. 24. Summary<br />Transformations need the following information:<br />Reflection A reflection line<br />Rotation A centre of rotation.<br /> An angle (usually given anti-clockwise)<br />Translation A column vector like <br />showing movement in the x- and y-directions<br /> .<br />Enlargement A scale factor and a centre of enlargement<br />
  25. 25. Images borrowed from CIMT’s MEP<br />