STRETCHES AND SHEARS

Stretches

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.

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

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’

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

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

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

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:

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

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’

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

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

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

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

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

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

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