STRETCHES AND SHEARS
Stretches
y
y
D
C
A
B
x
D’
C’
A’
x
B’
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.
y
D
C
A
B
x
y
D’
C’
A’
B’
x
2 Draw the image of ABCD after a stretch, stretch factor 3 with
the y axis invariant.
y
y
D
C
A
B
x
D’
C’
A’
x
B’
3 Draw the image of ABCD after a stretch, stretch factor 3 with
the x axis invariant.
y
y
D’
C’
D
C
A
B
x
A’
B’
x
The following diagram shows a stretch where the invariant line is not
the x or y axis.
y
x=1
8
C
A’B’ = 3 × AB
C’
So the stretch factor is 3.
6
4
The perpendicular distance of
each point from the line x = 1
has trebled.
2
So the invariant line is x = 1.
0
A
2
BA’
4
B’
6
8
10
x
If the scale factor is negative then the stretch is in the opposite direction.
y
B’
C’
C
B’C’ = 2 × BC and it has
been stretched in the
opposite direction.
B
8
So the stretch factor is −2.
6
A’
−6
−4
−2
4
The perpendicular distance of
each point from the y axis has
doubled.
2
So the invariant line is the y axis.
A
x
0
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:
shear factor =
distance moved by a point
perpendicular distance of point from the invariant line
y
y
D
C
A
B
D’
x
A’
C’
B’
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
So, shear factor   1
1
x
1 Draw the image of ABCD after a shear, shear factor 2 with
the x axis invariant.
y
y
D
A
D’
C
B
x
A’
B’
C’
x
2 Draw the image of ABCD after a shear, shear factor 1 with
the y axis invariant.
y
y
C’
D
C
A
B
D’
x
A’
B’
x
3 Draw the image of ABCD after a shear, shear factor 2 with
the y axis invariant.
y
y
C’
B’
D
C
A
B
D’
x
A’
x
4 Describe fully the single transformation that takes triangle A onto triangle B.
• shear
• invariant line is the x axis • shear factor is
8
2
4
y
8
4
A
2
0
B
4
x
2
4
6
8
5 Describe fully the single transformation that takes ABCD onto A’B’C’D’.
y
7
D
D’
C
C’
8
7
• shear
• invariant line is y = 2
7
• shear factor is  1
7
6
4
B A’
A
B’
y=2
2
0
2
4
6
8
10
x
6 Describe fully the single transformation that takes ABC onto A’B’C’.
y
8
8
B’
C’
• shear
• invariant line is the y axis
4 1
• shear factor is 
8 2
4
6
A’
C
B
4
A
2
0
2
4
6
8
10
x
7 Describe fully the single transformation that takes ABCD onto A’B’C’D’.
y
3
C D’
D
C’
8
3
• shear
6
y=6
4
A’
B’ A
B
2
0
2
4
6
8
10
x
• invariant line is y = 6
3
• shear factor is  1
3
8 Describe fully the single transformation that takes ABCD onto A’B’C’D’.
7
y
x=1
8
D
C
A
B
D’
• shear
note: this is a negative shear
7
6
• invariant line is x = 1
A’
4
• shear factor is 
C’
2
0
2
4
6
B’
8
10
x
7
 1
7