1. STRETCHES AND SHEARS

2. Stretches

3. A B C D A’ B’ C’ D’
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. A’ B’ C’ D’ A B C D

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

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

7. The following diagram shows a stretch where the invariant line is not the x or y axis.
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 2 4 6 8 10

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 −6 −4 −2 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. A B C D A’ B’ C’ D’
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. A B C D A’ B’ C’ D’

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

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

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

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

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

17. 7 Describe fully the single transformation that takes ABCD onto A’B’C’D’.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 2 4 6 8 10

18. 8 Describe fully the single transformation that takes ABCD onto A’B’C’D’.7 y
x = 1 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 2 4 6 8 10