22
Chapter
Transformations
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Reflections and line symmetry
Rotations and rotational
symmetry
Translations
Enlargements and reductions
Tessellations
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A
B
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Contents:
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IB MYP_1
404
TRANSFORMATIONS
(Chapter 22)
TRANSFORMATIONS
Translation, reflection, rotation, and enlargement are all transformations.
For example:
a translation
a reflection
mirror line
shift a particular distance in a particular direction
an enlargement
a rotation about O
O
When we perform a transformation, the original shape is called the object. The shape which
results from the transformation is called the image.
OPENING PROBLEM
Consider an equilateral triangle.
Things to think about:
² Can you draw a mirror line on an equilateral
triangle? The figure must fold onto itself along
that line, so it matches exactly.
How many of these mirror lines can you draw?
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² Can you find the centre of rotation of an equilateral triangle? The figure must rotate
about this point and fit exactly onto itself in less than one full turn.
How many times would an equilateral triangle fit onto itself in one full turn?
² Make a pattern using equilateral triangles so there are no gaps between the triangles
and the edges meet exactly.
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IB MYP_1
TRANSFORMATIONS
(Chapter 22)
405
CONGRUENT FIGURES
Two figures are congruent if they have exactly the same
size and shape.
We are congruent.
If one figure is cut out and it can be placed exactly on top
of the other, then these figures are congruent.
The image and the object for a translation, rotation, or
reflection are always congruent. The image and the object
for an enlargement are not congruent because they are not
the same size.
ACTIVITY 1
TRANSFORMING CATS
Following is a fabric pattern which features cats and pairs of cats. The rows
and columns have been numbered to identify each individual picture.
Col. 1
Col. 2
Col. 3
Col. 4
Col. 5
Col. 6
Row 1
Row 2
Row 3
Row 4
Row 5
What to do:
1 Start with row 1 and column 1 cat.
a Give the row and column numbers for translations of this cat.
b Give the row and column numbers for rotations of this cat.
c Give the row number in column 1 for an enlargement of this cat.
d Give the row numbers in column 3 for cats congruent to this cat.
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2 Start with row 1 column 2 cat.
a Give the row and column numbers for translations of this cat.
b Discuss why row 4 column 2 is not a rotation of this cat.
c Give the row and column numbers for rotations of this cat.
d Give the row number in column 1 for a reflection of this cat.
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IB MYP_1
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TRANSFORMATIONS
(Chapter 22)
3 Start with row 4 column 5 cat.
a Give the row numbers for any column 2
cats congruent to this one.
b Give the row and column numbers of any
cats that are a rotation of this cat.
c Give the row and column numbers for any
cats that are a reduction of this cat.
A reduction is the
opposite of an
enlargement. It makes
the figure smaller.
4 What is the transformation shown in the pair of cats?
5 Which two transformations are used to move the cats in
a row 1 column 1 to row 1 column 5
b row 2 column 4 to row 5 column 4
c row 2 column 5 to row 5 column 5
d row 3 column 6 to row 2 column 6?
A
REFLECTIONS AND LINE SYMMETRY
REFLECTIONS
object
To reflect an object in a mirror line we
DEMO
draw lines at right angles to the mirror
line which pass through key points on the
object. The images of these points are the
same distance away from the mirror line as the object
points, but on the opposite side of the mirror line.
image
mirror line
Example 1
Self Tutor
Draw the mirror image of:
object
mirror
image
EXERCISE 22A.1
1 Place a mirror on the mirror line shown using dashes, and observe the mirror image.
Draw the object and its mirror image in your work book.
a
b
c
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MAHS
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TRANSFORMATIONS
2
(Chapter 22)
407
a Draw the image of the following if a mirror was placed on the mirror line shown:
i
ii
iii
PRINTABLE
DIAGRAMS
b Check your answers to a using a mirror.
3 On grid paper, reflect the geometrical shape in the mirror line shown:
a
b
c
LINE SYMMETRY
A line of symmetry is a line along which a shape may be folded so that both parts of the
shape will match.
For example:
DEMO
fold line
line of symmetry
or
mirror line
If a mirror is placed along the line of symmetry, the reflection in the mirror will be exactly
the same as the half of the figure “behind” the mirror.
A shape has line symmetry if it has at least one line of symmetry.
Example 2
Self Tutor
For each of the following figures, draw all lines of symmetry:
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TRANSFORMATIONS
(Chapter 22)
a
c
b
4 lines of symmetry
3 lines of symmetry
no lines of symmetry
EXERCISE 22A.2
1 Copy the following figures and draw the lines of symmetry. Check your answers using
a mirror.
a
b
c
2
d
e
f
g
h
i
a Copy the following shapes and draw in all lines of symmetry.
i
ii
iii
iv
b Which of these figures has the most lines of symmetry?
3 How many lines of symmetry do these patterns have?
a
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a How many lines of symmetry can a triangle have? Draw all possible cases.
b How many lines of symmetry can a quadrilateral have? Draw all possible cases.
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TRANSFORMATIONS
ACTIVITY 2
(Chapter 22)
409
MAKING SYMMETRICAL SHAPES
You will need: paper, scissors, pencil, ink or paint
1 Take a piece of paper and fold it in
half.
2 Cut out a shape along the fold line.
3 Open out the sheet of paper and observe the shapes
revealed.
4 Record any observations about symmetry that you
notice.
5 Try the following:
a Fold the paper twice before cutting out your shape.
b Fold the paper three times before cutting out your shape.
In each case record your observations about the number of lines of symmetry.
6 Place a blob of ink or paint in the centre of a rectangular sheet of paper. Fold the
paper in half and press the two pieces together. Open the paper and comment on the
symmetry observed.
ink
fold line
fold
ink blot
7 Make symmetrical patterns by folding a piece of
paper a number of times and cutting out a shape.
How many folds would you need and what shape
would you need to cut out to get the result shown?
B
ROTATIONS AND
ROTATIONAL SYMMETRY
We are all familiar with things that rotate, such as the hands on a clock or the wheels of a
motorbike.
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DEMO
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TRANSFORMATIONS
(Chapter 22)
The point about which the hands of the clock, or the spokes of the wheel rotate, is called the
centre of rotation.
The angle through which the hands or the spokes turn is called the angle of rotation.
The globe of the world rotates about a line called the axis of rotation.
During a rotation, the distance of any point from the centre of rotation does not change.
A rotation is the turning of a shape or figure about a point and through a
given angle.
DEMO
For example:
180°
O
90°
90°
O
O
The figure is rotated
anticlockwise about O
through 90o .
The figure is rotated
anticlockwise about O
through 180o .
The figure is rotated
anticlockwise about O
through 90o .
You will notice that under a rotation, the figure does not change in size or shape.
In mathematics we rotate in an anticlockwise direction unless we are told otherwise.
You should remember that 90o is a
360o is a full turn.
1
4 -turn,
180o is a
1
2 -turn,
270o is a
Example 3
3
4 -turn,
and
Self Tutor
Rotate the given figures about O through the angle indicated:
a
b
c
180°
90°
270°
O
O
O
a
b
c
180°
270°
90°
O
O
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O
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IB MYP_1
TRANSFORMATIONS
(Chapter 22)
411
EXERCISE 22B.1
1 Consider the rotations of
which follow:
O
A
B
C
O
D
O
O
O
Which of A, B, C, or D is a rotation of the object through:
a 180o
b 360o
c 90o
d 270o ?
2 Copy and rotate each of the following shapes about the centre of rotation O, for the
number of degrees shown. You could use tracing paper to help you.
a
b
c
O
O
90°
d
O
180°
f
e
270°
O
PRINTABLE
WORKSHEET
O
360°
O
90°
g
270°
h
i
O
180°
O
90°
3 Rotate about O through the angle given:
b 180o
a 90o
90°
c 270o
O
O
O
d 180o
e 90o
DEMO
f 270o
O
O
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TRANSFORMATIONS
(Chapter 22)
ROTATIONAL SYMMETRY
A full rotation does not
mean that a shape has
rotational symmetry.
Every shape fits exactly
onto itself after a
rotation of 360o.
A shape has rotational symmetry
if it can be fitted onto itself by
turning it through an angle of less
than 360o, or one full turn.
O
The centre of rotational
symmetry is the point about
which a shape can be rotated onto
itself.
The ‘windmill’ shown will fit onto itself every time it is
turned about O through 90o. O is the centre of
rotational symmetry.
This fabric pattern also shows rotational symmetry.
DEMO
If a figure has more than one line of symmetry then it will also have rotational symmetry.
The centre of rotational symmetry will be the point where the lines of symmetry meet.
Example 4
Self Tutor
For the following figures, find the centre of rotational symmetry.
a
b
a
b
c
c
O
O
O
centre is O
centre is O
centre is O
THE ORDER OF ROTATIONAL SYMMETRY
DEMO
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The order of rotational symmetry is the number of times a figure
maps onto itself during one complete turn about the centre.
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IB MYP_1
TRANSFORMATIONS
413
(Chapter 22)
For example:
A
B
C
D
180°
rotation
D
C
A
B
D
C
360°
rotation
B
A
centre of rotation
DEMO
The rectangle has order of rotational symmetry of 2 since it moves back
to its original position under rotations of 180o and 360o:
Click on the icon to find the order of rotational symmetry for an
equilateral triangle.
EXERCISE 22B.2
1 For each of the following shapes, find the centre of rotational symmetry:
a
b
c
2 For each of the following shapes find the order of rotational symmetry. You may use
tracing paper to help you.
a
b
c
d
e
f
g
h
3 Design your own shape which has order of rotational symmetry of:
a 2
b 3
c 4
ACTIVITY 3
d 6
USING TECHNOLOGY TO ROTATE
In this activity we use a computer package to construct a shape
that has rotational symmetry.
ROTATING
FIGURES
What to do:
1 Click on the icon to load the software.
2 From the menu, choose an angle to rotate through.
3 Make a simple design in the sector which appears, and colour it.
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to see your creation.
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4 Press finish
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TRANSFORMATIONS
(Chapter 22)
C
TRANSLATIONS
A translation of a figure occurs when every point on the figure is moved the same distance
in the same direction.
Under a translation the original figure and its image are congruent.
DEMO
4
A
object
3
In the translation shown, the original figure has
been translated 4 units right and 3 units down
to give the image.
A'
image
EXERCISE 22C
1 Describe each of the following translations:
a
b
c
object
image
object
image
image
object
d
e
object
object
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B to A
C to B
C to A
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2 For the given figures,
translation from:
a A to B
b
c B to C
d
e A to C
f
image
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IB MYP_1
TRANSFORMATIONS
(Chapter 22)
3 Copy onto grid paper and translate using the given directions:
a 3 right, 4 down
D
b 6 left, 4 up
415
DEMO
c 2 right, 5 up
ENLARGEMENTS AND REDUCTIONS
We are all familiar with enlargements in the form of photographs or looking through a
microscope or telescope. Plans and maps are examples of reductions. The size of the image
has been reduced but the proportions are the same as the original. Most photocopiers can
perform enlargements and reductions.
The following design shows several enlargements:
In any enlargement or reduction, we multiply the lengths in the object by the scale factor
to get the lengths in the image.
Look at the figures in the grid below:
A
B
C
enlargement
scale factor 2
D
DEMO
enlargement
scale factor 3
For the enlargement with scale factor 2, lengths have been doubled.
For the enlarement with scale factor 3, lengths have been trebled.
If shape B is reduced to shape A, the lengths are halved and the scale factor is 12 .
If shape D is reduced to shape A, the lengths are quartered and the scale factor is 14 .
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² greater than 1 for an enlargement
² less than 1 for a reduction.
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A scale factor is:
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IB MYP_1
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TRANSFORMATIONS
(Chapter 22)
ACTIVITY 4
ENLARGEMENT BY GRIDS
You will need: Paper, pencil, ruler
What to do:
1 Copy the picture alongside.
2 Draw a grid 5 mm by 5 mm over the top
of the dog as shown alongside:
3 Draw a grid 10 mm by 10 mm alongside the grid
already drawn.
4 Copy the dog from the smaller grid onto the larger
grid. To do this accurately, start by transferring points
where the drawing crosses the grid lines. Then join
these points and finish the picture.
5 Use this method to change the size of
other pictures. You may like to try
making the picture smaller as well as
larger, by making your new grid smaller
than the original.
PRINTABLE
WORKSHEET
Example 5
Self Tutor
Enlarge
using a scale factor of:
a 2
a
1
2
b
b
1
2
2
4
EXERCISE 22D
1 In the following diagrams, A has been enlarged to B. Find the scale factor.
a
b
c
A
B
A
B
B
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TRANSFORMATIONS
(Chapter 22)
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2 In the following diagrams, A has been reduced to B. Find the scale factor.
a
b
c
B
B
A
A
B
A
3 Enlarge or reduce the following objects by the scale factor given:
a
b
scale factor 2
scale factor Qw_
scale factor 3
d
PRINTABLE
WORKSHEET
c
e
f
scale factor Qe_
scale factor 4
scale factor 2
4 Find the scale factor when A is transformed to B:
a
b
A
c
A
A
B
B
B
5 For each grid in 4, write down the scale factor which transforms B into A.
E
TESSELLATIONS
A tessellation is a pattern made using figures of the same shape and size. They must cover
an area without leaving any gaps.
The photograph alongside shows a tessellation of bricks
used to pave a footpath.
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Tessellations are also found in carpets, wall tiles, floor
tiles, weaving and wall paper.
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TRANSFORMATIONS
(Chapter 22)
This brick design is not a tessellation as it is
constructed from two different brick sizes.
The following tile patterns are all tessellations:
For a tessellation
the shapes must
fit together with
no gaps.
ACTIVITY 5
PAVING BRICKS
What to do:
1 Using the “2 £ 1” rectangle
, form at least two different
tessellation patterns. One example is:
2 Repeat 1 using a “3 £ 1” rectangle
.
Example 6
Self Tutor
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Draw tessellations of the following shapes.
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IB MYP_1
TRANSFORMATIONS
(Chapter 22)
419
EXERCISE 22E
1 Draw tessellations using the following shapes:
a
b
c
2 Draw tessellations using the following shapes:
a
b
c
ACTIVITY 6
CREATING TESSELLATIONS
What to do:
Follow these steps to create your own tessellating pattern.
Step 1: Draw a square.
Step 2: Cut a piece from one side and ‘glue’
it onto the opposite side.
Step 3: Rub out any unwanted
lines and add features.
Step 4: Photocopy this several times and cut
out each face. Combine them to
form a tessellation.
Make your own tessellation pattern and produce a full page pattern with 3 cm by 3 cm
tiles. Be creative and colourful. You could use a computer drawing package to do this
activity.
DISCUSSION
IN GOOD SHAPE
1 Research the shape of the cells in a beehive.
Explain why they are that shape.
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2 Look at the shapes of paving blocks.
Explain what advantage some shapes have
over others. When building walls, what are
the advantages of rectangular bricks over
square bricks?
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TRANSFORMATIONS (Chapter 22)
ACTIVITY 7
COMPUTER TRANSFORMATIONS
What to do:
1 Pick a shape and learn how to translate, reflect and
rotate it.
TESSELLATIONS
BY COMPUTER
2 Create a tessellation on your screen and colour it.
3 Print your final masterpiece.
ACTIVITY 8
DISTORTION TRANSFORMATIONS
In this activity we copy pictures onto unusual graph paper to produce
distortions of the original diagram. For example,
on
becomes
What to do:
PRINTABLE
GRIDS
1 On ordinary squared paper draw a picture of your own choosing.
2 Redraw your picture on different shaped graph paper. For example:
KEY WORDS USED IN THIS CHAPTER
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axis of rotation
enlargement
mirror line
rotation
tessellation
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²
²
²
²
²
angle of rotation
congruent
line of symmetry
reflection
scale factor
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²
²
²
²
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centre of rotation
image
object
rotational symmetry
translation
IB MYP_1
TRANSFORMATIONS (Chapter 22)
421
REVIEW SET 22A
1 Draw the mirror image of:
a
b
mirror line
2 Draw the lines of symmetry for:
mirror line
3 Rotate the given figure about O
through 90o anticlockwise.
O
4 For the given shape:
a draw the lines of symmetry
b state the order of rotational symmetry.
5
a Translate the figure three units
left and one unit up.
b Enlarge the figure with
scale factor 2.
6
In the diagram A has been reduced to B.
Find the scale factor.
A
B
7 Draw tessellations of the following shapes.
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IB MYP_1
422
TRANSFORMATIONS (Chapter 22)
REVIEW SET 22B
1 Draw the mirror image of:
a
b
mirror line
mirror line
2 Draw the lines of symmetry for:
3
a
b
a Rotate the figure shown through
90o anticlockwise about O.
b Enlarge the figure with scale
factor 13 .
O
4
b Rotate the given figure 180o
about O.
a Find the order of rotational
symmetry for:
O
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Y:\HAESE\IB_MYP1\IB_MYP1_22\422IB_MYP1_22.CDR Friday, 18 July 2008 3:40:50 PM PETER
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6 Draw a tessellation using the given
shape.
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5 Translate the given figure one unit
to the right and 3 units down.
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IB MYP_1