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STRETCHES AND SHEARS

STRETCHES AND SHEARS. Stretches. D. D’. C. C’. B. B’. A. A’. In this example ABCD has been stretched to give A’B’C’D’. The points on the y axis have not moved, so the y axis (or x = 0) is called the invariant line.

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STRETCHES AND SHEARS

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  1. STRETCHES AND SHEARS

  2. Stretches

  3. D D’ C C’ B B’ A A’ In this example ABCD has been stretched to give A’B’C’D’. The points on the y axis have not moved, so the y axis (or x = 0) is called the invariant line. The perpendicular distance of each point from the invariant line has doubled, so the stretch factor is 2.

  4. 1 Draw the image of ABCD after a stretch, stretch factor 2 with the x axis invariant. D’ C’ D C B B’ A A’

  5. 2 Draw the image of ABCD after a stretch, stretch factor 3 with the y axis invariant. D D’ C C’ B B’ A A’

  6. D’ C’ 3 Draw the image of ABCD after a stretch, stretch factor 3 with the x axis invariant. D C B B’ A A’

  7. The following diagram shows a stretch where the invariant line is not the x or y axis. y x = 1 A’B’ = 3 × AB C’ C 8 So the stretch factor is 3. 6 The perpendicular distance of each point from the line x = 1 has trebled. 4 So the invariant line is x = 1. 2 A B A’ B’ x 0 4 2 6 10 8

  8. If the scale factor is negative then the stretch is in the opposite direction. y B’C’ = 2 × BC and it has been stretched in the opposite direction. B’ C’ C B 8 So the stretch factor is −2. 6 The perpendicular distance of each point from the y axis has doubled. 4 So the invariant line is the y axis. 2 A’ A x −4 −6 −2 0 2 4

  9. Shears In a shear, all the points on an object move parallel to a fixed line (called the invariant line). A shear does not change the area of a shape. To calculate the distance moved by a point use:

  10. D’ C’ D C B B’ A A’ In this example ABCD has been sheared to give A’B’C’D’. The points on the x axis have not moved, so the x axis (or y = 0) is called the invariant line. DD’ = 1 and distance of D from the invariant line = 1

  11. 1 Draw the image of ABCD after a shear, shear factor 2 with the x axis invariant. D’ C’ D C B B’ A A’

  12. 2 Draw the image of ABCD after a shear, shear factor 1 with the y axis invariant. C’ D’ D C B’ B A A’

  13. C’ 3 Draw the image of ABCD after a shear, shear factor 2 with the y axis invariant. B’ D’ D C B A A’

  14. 4 Describe fully the single transformation that takes triangle A onto triangle B. • shear factor is • invariant line is the x axis • shear 8 4 A B 4 2 0 2 4 8 6

  15. 5 Describe fully the single transformation that takes ABCD onto A’B’C’D’. y 7 D C D’ C’ 8 • shear • invariant line is y = 2 6 7 • shear factor is 4 A B A’ B’ y = 2 2 x 0 4 2 6 10 8

  16. 6 Describe fully the single transformation that takes ABC onto A’B’C’. y 8 8 B’ • shear C’ • invariant line is they axis 4 6 • shear factor is A’ C B 4 A 2 x 0 4 2 6 10 8

  17. 7 Describe fully the single transformation that takes ABCD onto A’B’C’D’. y 3 D C D’ C’ 8 3 • shear 6 y = 6 • invariant line is y = 6 • shear factor is 4 A’ B’ A B 2 x 0 4 2 6 10 8

  18. 8 Describe fully the single transformation that takes ABCD onto A’B’C’D’. 7 x = 1 y D C 8 A B • shear D’ note: this is a negative shear 7 6 • invariant line is x = 1 A’ 4 • shear factor is C’ 2 B’ x 0 4 2 6 10 8

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