1. Unit 7.5: Shear A transformation that changes the shape but not the size of the figure.

2. Shear The transformation consists of: The invariant line (x-axis or y-axis or any line parallel to the x- and y-axes) Shear factor, k Shear factor, k = displacement of image point from object point displacement of object point from L (invariant line)

3. Find the shear factor, k given the figures of object and image under shear with x-axis as the invariant line. All points move a distance parallel to the invariant line, except points which are on the invariant line (x-axis) POINTS ON INVARIANT LINE DO NOT MOVE !! 4 8

4. Shear this triangle by the shear factor 1. The line AB is the invariant line Shear factor = C B A C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line

5. Shear this triangle by the shear factor -1. The line AB is the invariant line Shear factor = C BA C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line

6. Shear this triangle by the shear factor 1. The x-axis is the invariant line Shear factor = C B A C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line A’ B’

7. Shear this triangle by the shear factor 2. The x-axis is the invariant line Shear factor = C BA C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line

8. Shear this triangle by the shear factor 1. The line y = 4 is the invariant line C B A C’ Invariant line A’B’ If invariant line is x-axis or parallel to x-axis: Above invariant line If k = +ve (shear to the right) If k = -ve (shear to the left) Below invariant line If k = +ve (shear to the left) If k = -ve (shear to the right)

9. Shear this triangle by the shear factor 1½. The y-axis is the invariant line Shear factor = C B A C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line

10. Shear this triangle by the shear factor -2. The y-axis is the invariant line Shear factor = C B A C’ Invariant line Distance a point moves due to shear Perpendicular distance of point from the invariant line A’ B’

11. Shear this triangle by the shear factor 1. The line x = 4 is the invariant line C B A C’ Invariant line A’ B’ If invariant line is y-axis or parallel to y-axis: Right of invariant line If k = +ve (shear upwards) If k = -ve (shear downwards) Left of invariant line If k = +ve (shear downwards) If k = -ve (shear upwards)

12. Describe fully the transformation under shear. Important points required to describe a shear: the word ‘shear’ invariant line shear factor, k

13. 2 4 6 8 10 2 4 6 y x A Describe fully the transformation that takes triangle A onto triangle B B shear invariant line is shear factor is the x axis 8 = 2 4 8 4

14. 2 4 6 8 10 2 4 6 y x A B shearinvariant line isshear factor is the y axis 4 = 1 8 2 8 4 Describe fully the transformation that takes triangle A onto triangle B

15. 2 4 6 8 10 2 4 6 y x A B Describe fully the transformation that takes triangle A onto triangle B shear invariant line is shear factor is y = 1 6 = 1 6 6 6