STRETCHES AND SHEARS

Stretches

x

y

x

y

A B

CD

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.

x

y

x

y

A B

CD

A’ B’

C’D’

1 Draw the image of ABCD after a stretch, stretch factor 2 with the x axis invariant.

x

y

x

y

A B

CD

A’ B’

C’D’

2 Draw the image of ABCD after a stretch, stretch factor 3 with the y axis invariant.

x

y

x

y

A B

CD

A’ B’

C’D’

3 Draw the image of ABCD after a stretch, stretch factor 3 with the x axis invariant.

The following diagram shows a stretch where the invariant line is not the x or y axis.

x

y

2 4 6 80 10

2

4

6

8

A B

C

A’ B’

C’A’B’ = 3 × AB

So the stretch factor is 3.

The perpendicular distance of each point from the line x = 1 has trebled.

So the invariant line is x = 1.

x = 1

If the scale factor is negative then the stretch is in the opposite direction.

x

y

2 4−2−4 0

2

4

6

8

A

BC

A’

B’ C’ B’C’ = 2 × BC and it has been stretched in the opposite direction.

So the stretch factor is −2.

The perpendicular distance of each point from the y axis has doubled.

So the invariant line is the y axis.

−6

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.

Shears

shear factor =

distance moved by a point

perpendicular distance of point from the invariant line

To calculate the distance moved by a point use:

x

y

x

y

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.

A B

CD

A’ B’

C’D’

DD’ = 1 and distance of D from the invariant line = 1

So, shear factor

1

11

x

y

x

y

A B

CD

A’ B’

C’D’

1 Draw the image of ABCD after a shear, shear factor 2 with the x axis invariant.

x

y

x

y

2 Draw the image of ABCD after a shear, shear factor 1 with the y axis invariant.

A B

CD

A’

B’

C’

D’

x

y

x

y

A B

CD

A’

B’

C’

D’

3 Draw the image of ABCD after a shear, shear factor 2 with the y axis invariant.

2 4 6 8

2

4

A

4 Describe fully the single transformation that takes triangle A onto triangle B.

B

• shear • invariant line is the x axis • shear factor is

8

4

y

x0

8

42

D

5 Describe fully the single transformation that takes ABCD onto A’B’C’D’.

x

y

2 4 6 80 10

2

4

6

8

A B

C

A’ B’

C’D’

• shear

• invariant line is y = 2

• shear factor is

7

71

y = 2

7

7

6 Describe fully the single transformation that takes ABC onto A’B’C’.

x

y

2 4 6 80 10

2

4

6

8

A

BCA’

B’C’ • shear

• invariant line is the y axis

• shear factor is

4

8

1

2

8

4

D

7 Describe fully the single transformation that takes ABCD onto A’B’C’D’.

x

y

2 4 6 80 10

2

4

6

8

A B

C

A’ B’

C’D’

• shear

• invariant line is y = 6

• shear factor is

3

31

y = 6

3

3

D

8 Describe fully the single transformation that takes ABCD onto A’B’C’D’.

x

y

2 4 6 80 10

2

4

6

8 A B

C

A’

B’

C’

D’

• shear

• invariant line is x = 1

• shear factor is

7

7 1

x = 1

7

7

note: this is a negative shear