18
ES
Ch apter
PA
G
Measurement and Geometry
N
AL
Transformations and
symmetry
FI
Consider the triangle T drawn below. Each of the triangles T1, T2, T3 is obtained
from the triangle T by moving it in different ways.
T
T1
Diagram 1
T
T2
T
T3
Diagram 2 Diagram 3
When the figure is moved, we call this movement a transformation. The
resulting figure is called the image of the original figure.
continued over page
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441
We will be dealing with only three types of transformation in this chapter:
1 translations (diagram 1)
2 rotations (diagram 2)
3 reflections (diagram 3).
Note that none of these transformations changes the size or shape of an object.
You will have seen these three transformations in many applications on your computer.
All of these transformations satisfy two important properties.
Properties of translations, rotations and reflections
ES
When a figure is translated, rotated or reflected:
• intervals move to intervals of the same length
• angles move to angles of the same size.
Translations
N
AL
18 A
PA
G
Many of these exercises will require grid paper. The work on rotations will require some work
on polar graph paper, also called circular graph paper. You can download samples of this from
your Interactive Textbook. Carefully drawn diagrams are essential for understanding.
Shifting a figure in the plane without turning it is called a translation. To describe a translation, it is
enough to say how far left or right, and how far up or down, the figure is moved.
In the diagram opposite, triangle ABC has been translated to a new position on
the page, and the new triangle has been labelled A′B′C ′.
A′
The image of a point is usually written with a dash attached. Thus the image
of A is written A′ (read as ‘A prime’ or ‘A dash’). The image of Δ ABC is thus
Δ A ′B ′C ′.
B
FI
You can see that triangle ABC has been shifted 2 units right and 1 unit up. Point
A has moved to A′, point B to B′ and point C to C′.
A
B′
C′
C
You can see that our translation has changed neither the side lengths nor the angles of the triangle, in
agreement with the properties listed above.
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1 8 A T r a n s l at i o n s
Example 1
Describe how the triangle in the diagram
opposite has been translated.
A
A′
C
B
B′
C′
Solution
ES
The figure has been translated 3 units left and 1 unit down. (Note that 1 unit down
followed by 3 units left is the same translation.)
G
Triangle ABC is shown by a blue line while the image triangle A′B′C′ is shown with a red line, a
convention we will stay with throughout the chapter.
PA
Example 2
N
AL
Translate Δ XYZ in the diagram opposite
2 units down and 3 units left.
X
Y
Z
Solution
FI
We only need to shift the vertices of this
triangle. We then join them up to give the
translated figure and label it X′Y′Z′.
X
X’
Y
Z
Y’
Z’
Notice that in each of the examples above, every point of the triangle has moved. This is a special
property of all translations, apart from the identity translation.
The translation that translates a figure 0 units right and 0 units up is called the identity translation.
The identity translation leaves all points fixed.
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1 8 A T r a n s l at i o n s
Properties of translations
When a non-identity translation is applied:
• every interval is translated to an interval of equal length
• every angle is translated to an angle of equal size
• every point moves; that is, there are no fixed points.
Exercise 18A
1 Describe the translation shown in each diagram below. You need not copy the diagrams.
a
A′
b
ES
Example 1
c
A′
A
A
G
A′
B′
B
d
PA
A
e
A
A’
A
Example 2
A′
B′ C
N
AL
B
C
B
C′
B′
f
A
A′
C
B
C′
B′
C′
2 In each part below, a figure is shown and a translation described. Copy each figure and
draw its image after the translation. Remember to label your images using dashes.
b 3 units right, 1 unit up
A
FI
a 4 units up, 3 units right
A
c 2 units up, 1 unit left
B
d 2 units right, 1 unit down
A
A
B
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1 8 A T r a n s l at i o n s
3 In the diagram below, Δ X ′ Y ′ Z ′ is the image under translation of Δ XYZ. Copy the diagram
onto your graph paper.
X
X′
Y
Z
Z′
Y′
a Measure angles XYZ and X′Y′Z′ with your protractor. What do you notice?
ES
b Write down the angle in the image that is the same size as:
i ∠XZY
ii ∠ZXY
(Check with a protractor.)
G
c Accurately measure the lengths of the intervals XY and X′Y′. Comment.
d Calculate the area, in square units, of Δ XYZ and Δ X ′ Y ′ Z ′ . Comment.
PA
4 In the diagram below, Δ A ′ B′ C ′ is the image under a translation of Δ ABC. Copy the
diagram onto your graph paper.
A
N
AL
A′
B
L
C
B′
C′
a Mark the image of L under the same translation; label it L′.
b Describe the translation in words.
FI
c Are the intervals AB and A′B′ parallel?
d Write down the interval in the image that is parallel to:
i AC
ii AL
e Write down the interval in the image that is parallel to and the same length as:
i BL
ii CA
f Are there any points in the original figure that have not moved under the translation?
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445
18 B
Rotations
A rotation turns a figure about a fixed point, called the centre of rotation. The centre of rotation is
usually labelled by the letter O.
A rotation is specified by:
• the centre of rotation O
• the angle of rotation
• the direction of rotation (clockwise or anticlockwise).
ES
In the first diagram below, the point A is rotated through 120° clockwise about O. In the second
diagram, it is rotated through 60° anticlockwise about O.
A’
60°
G
A′
O
O
120°
A
A
PA
The examples above demonstrate the following special properties of any rotation.
The identity rotation is the rotation that rotates a figure through 0° and leaves every point fixed.
Properties of rotations
For any non-identity rotation:
N
AL
• every interval is rotated to an interval of equal length
• every angle is rotated to an angle of equal size
• there is only one fixed point – the centre of rotation
• the distance of a point from the centre of rotation does not change.
FI
To help us when working with rotations, we may use special graph paper called polar or circular
graph paper. This is available in your Interactive Textbook. We will use two types of such paper,
examples of which can be seen in the next two diagrams. The first is marked with radial lines spaced
at 30° apart, while in the second the lines are spaced at 45°. The centre of rotation O is always the
centre of the graph paper and is called the origin.
In this graph paper, lines radiate out from O at 30°
angles. The diagram is a bit like a clock face, with
the lines pointing towards the hours.
446
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1 8 B R o tat i o n s
The second diagram is like a compass,
with the lines 45° apart pointing to
N, NE, E, SE, S, SW, W and NW.
O
The next example shows how the image of an interval under a rotation may be found by rotating its
endpoints and joining their images.
Example 3
E
ES
Rotate the interval ED in the diagram by 120° in
a clockwise direction about O.
D
G
O
Solution
N
AL
PA
To rotate the points D and E by 4 × 30° = 120°
clockwise, shift each point four ‘hours’ around
the ‘clock’. Each point stays the same distance
from the centre of rotation; that is, it stays on the
same circle. The points D and E are rotated and
then joined up to form the image internal D′E′.
E
D
O
E′
D′
Example 4
Rotate Δ XYZ in the diagram by 90°
anticlockwise about O.
X
O
Z
FI
Y
Solution
A rotation of 90° anticlockwise moves every
point though two divisions on the ‘compass’.
To move the triangle XYZ, we rotate the three
corners and join them up to form the image
triangle X′Y′Z′.
X
O
Z
Y′
Y
X′
Z′
You can see that lengths of intervals do not change.
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447
1 8 B R o tat i o n s
In the following example, we will be concerned with rotating figures 90°, 180° or 270°, so a set
square is ideal to construct these angles.
Example 5
Rotate the line AB by 90° clockwise about O.
B
A
ES
O
Solution
G
We only need to rotate the endpoints A and B. The image of the segment AB is obtained by
joining A′ and B′. This is explained in the following diagrams.
B
A
PA
A′
O
N
AL
To rotate A we have drawn lines at 90°, starting with OA. A set square and a ruler can then
help us rotate A around O to A′, keeping it always the same distance from O, since A and A′
are an equal distance from O.
B
A
A′
O
B′
FI
Repeat this process to rotate B around O to B′.
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1 8 B R o tat i o n s
Exercise 18B
Example
3, 4
Use the appropriate type of polar graph paper in Question 1 and mark the points and
figures carefully.
1 Rotate each figure about the centre O through the given angle. Remember to label your
images with dashed letters.
a Rotate 135° anticlockwise.
b Rotate 90° clockwise.
N
O
M
X
Y
G
d Rotate 30° clockwise.
PA
Z
O
Q
P
R
f Rotate 30° anticlockwise.
N
AL
e Rotate 120° anticlockwise.
A
O
O
2 Rotate each figure through the given angle about the given centre O. Remember to label
your images with dashed letters.
FI
Example 5
B
C
c Rotate 90° anticlockwise.
O
ES
O
A
a Rotate 90° anticlockwise.
b Rotate 90° anticlockwise.
O
O
A
A
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1 8 B R o tat i o n s
c Rotate 90° clockwise.
d Rotate 90° clockwise.
A
A
O
O
B
B
e Rotate 270° clockwise.
f Rotate 270° anticlockwise.
O
ES
O
g Rotate 180° clockwise.
O
C
O
j Rotate 180° anticlockwise.
B
A
A
PA
i Rotate 180° anticlockwise.
B
B
A
G
B
h Rotate 180° clockwise.
A
O
C
O
N
AL
C
k Rotate 90° clockwise.
D
l Rotate 90° clockwise.
A
O
FI
O
A
3 Consider Question 2 above. Were any points fixed under the rotations, or did they all move?
4 Consider the following diagram of a house.
a Draw the image of the house after a rotation of 90°
anticlockwise about O. Label it A′B′C′D′E′.
D
b AE and BC are parallel. What do you notice about
A′E′ and B′C′?
c Find the area of ABCE and A′B′C′E′ in square units.
Comment.
O
450
E
C
A
B
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18 C
Reflections
G
ES
A reflection is a transformation that flips a figure about a line. This line is called the axis of
reflection. A good way to understand this is to suppose that you have a book with clear plastic pages
and a figure drawn, as in the picture below. If the page is turned, the triangle is flipped over. We say
it has been reflected; in this case, the axis of reflection is the binding of the book.
Our later examples will look more like the diagram opposite. The axis of
reflection has also been drawn.
PA
A
The image of each point can be determined in the following way.
A’
N
AL
To reflect A in the axis of reflection, draw a line at right angles to the axis of
reflection. Then use a ruler or compasses to mark A′ at the same distance as A,
but on the other side. This is best done as a construction using a set square and
compasses.
A
A
A’
FI
As was the case for rotation, we find the image of a triangle under a reflection by reflecting the
vertices and joining the image vertices.
Example 6
Reflect ΔABC in the given axis of
reflection shown.
A
B
C
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1 8 C Re f l ect i o n s
Solution
B
B’
C
A′
A
A’
A
B
B′
C
C’
Join them up to get the image triangle.
ES
Reflect each of the vertices.
C′
Here is a list of special properties of reflections. The second property can be seen in the previous
example.
G
Properties of reflections
When a reflection is applied:
PA
• all points on the axis of reflection are fixed points; the only fixed points are those on the
axis of reflection
• if the points A, B, C, … are in a clockwise order, then the points A′, B′, C′, … will be in
anticlockwise order, and vice versa
• if a reflection is applied twice, all figures are returned to their original position
• every interval is reflected to an interval of equal length
N
AL
• every angle is reflected to an angle of equal size.
Exercise 18C
1 Reflect each figure in the given axis of reflection. (In each diagram, the axis of reflection is
marked with an arrow at each end.) Remember to label your images in the proper way.
FI
Example 6
a
b
c
X
A
A
Z
B
B
Y
C
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1 8 C Re f l ect i o n s
e
d
f
X
A
A
Z
B
Y
B
C
g
h
i
X
A
Z
ES
A
B
Y
B
k
PA
j
G
C
n
o
N
AL
m
l
FI
2 This question involves two successive reflections. First reflect the triangle in the axis of
reflection labelled m, and then in the axis of reflection labelled n.
a
m
b
n
m
n
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453
18 D
Combinations of
transformations
Here is a summary of what we have discovered so far about translations, rotations and reflections:
• Intervals move to intervals of the same length.
• Angles move to angles of the same size.
• Pairs of parallel lines move to pairs of parallel lines.
Defined by
Fixed points
Translation
Vertical and horizontal shift
None
Rotation
Centre, angle and direction
Reflection
Axis of reflection
ES
Transformation
The centre of rotation
Points on the axis of reflection
G
Reflection is the only one of the three transformations that reverses the order of vertices. Thus
if a triangle is labelled ABC in clockwise order, then the vertices of Δ A′B′C′ will appear in
anticlockwise order after reflection.
A′
PA
A
B
N
AL
C
B′
C′
This is different from the case when a translation or rotation is applied to Δ ABC with vertices in
clockwise order. The vertices of Δ A′B′C′ will also be in clockwise order, as shown below.
FI
We can combine two or more transformations by applying them one after another, as in the
examples below.
Example 7
Translate Δ ABC 2 units right and 1 unit up. Then
translate the image 3 units left and 2 units up. Describe
a single translation that has the same effect.
C
A
454
B
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1 8 D C o mb i n at i o n s o f t r a n s f o r mat i o n s
Solution
The successive transformations are shown in the
diagram opposite. The combined translation is the
same as a translation 1 unit left and 3 units up.
C″
C′
C B″
A″
A′
A
B′
ES
Example 8
B
Rotate the point A by 45° and then by a further 90°, both rotations
anticlockwise about O. Describe a single rotation that has the
same effect as the combination of these two rotations.
G
PA
Solution
N
AL
The first rotation moves A to A′ (see the diagram
opposite). The second rotation then moves A′ to A″.
This combined transformation is the same as a single
rotation of 135° anticlockwise about O.
A
O
135°
A′
45°
A″
A 0°
O
Example 9
FI
Describe two successive transformations that will have
the combined effect of moving ΔABC onto Δ A′B′C′ as
shown opposite.
A′
C
A
B
B′
C′
Solution
There are many possible answers. One possible answer is:
• first translate Δ ABC so that A moves to A′; that is, 4 units right and 3 units up
• then reflect the triangle in A′B′.
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1 8 D C o mb i n at i o n s o f t r a n s f o r mat i o n s
Example 10
B
ℓ
C
A
ES
a Reflect triangle ABC in the line ℓ and then translate the image 5 units to the right.
b Translate triangle ABC 5 units to the right and then reflect the image in the line ℓ.
B
A
b
ℓ
C
B
B″
C′
A′ C″
ℓ
B′
A″
N
AL
B″
B′
PA
a
G
Solution
A
C″
C
A″ A′
C′
FI
Note that in parts a and b a different result is obtained. The final image depends on the order in
which the transformations are taken.
The following exercise involves no new transformations, but uses combinations of the three
transformations we have learned about. If you get stuck on any step, look back at the earlier exercise
dealing with the relevant kind of transformation.
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1 8 D C o mb i n at i o n s o f t r a n s f o r mat i o n s
Exercise 18D
Example 7
1 In each part perform the two transformations successively. Then find a single translation
that has the same effect.
a Translate 1 right, 2 up,
then 3 right, 1 up.
bTranslate 3 down, then 2 right.
ES
A
d Shift 1 right, 3 up, then 2 down, 2 right.
Example 8
PA
G
c Shift 3 right, 2 down, then 2 up.
2 Carry out the first rotation given for each figure, draw the image, and then carry out the
second rotation. Find a single rotation that will have the same effect.
a Rotate 60° anticlockwise, then 30° anticlockwise.
b Rotate 30° anticlockwise, then 60° clockwise.
N
AL
c Rotate 120° anticlockwise, then 90° anticlockwise.
Copy the first diagram below.
FI
3 a i ii Reflect the triangle in m, then reflect the image in n.
iii Now reflect the original triangle in n, then reflect the image in m.
iv Are your answers to parts ii and iii the same?
b Repeat the steps i to iv for the second diagram.
n
n
m
m
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Example 9
4 In each part, describe two successive transformations that will have the combined effect of
moving Δ ABC onto Δ A″B″C″.
a
b
B″
A
B″
A″
A″
A
C″
B
C″
B
C
c
C
d
ES
C″
B
C
B″
A″
C, A″
B″
B
A
C″
5 a Reflect triangle ABC in the line ℓ and then translate the image 6 units to the right.
PA
Example 10
G
A
b Translate triangle ABC 6 units to the right and then reflect the image in the line ℓ.
c Translate triangle ABC 6 units up and the reflect the image in the line ℓ.
d Reflect triangle ABC in the line ℓ and then translate the image 6 units up.
N
AL
ℓ
B
A
C
FI
6 a Reflect triangle ABC in the line ℓ and then translate the image 6 units to the right.
b Translate triangle ABC to the right 6 units and then reflect in the line ℓ.
B
A
C
ℓ
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7 a Rotate triangle ABC about point O in an anticlockwise direction by 90° and then reflect
the image in the line ℓ.
b Reflect triangle ABC in the line ℓ and then rotate
the image about point O in an anticlockwise
direction by 90°.
B
A
18 E
C
O
ℓ
ES
Transformations in
the Cartesian plane
G
In this section, transformations are described using coordinates.
PA
Example 11
Find the coordinates of the image of the point (1, 3) under:
a translation of 1 unit to the right and 3 units up
a translation of 3 units to the left and 4 units down
a clockwise rotation of 90° about the origin
an anticlockwise rotation of 90° about the origin
a reflection in the x-axis
a reflection in the y-axis
a rotation of 180° about the origin
N
AL
a
b
c
d
e
f
g
Solution
y
FI
A(2, 6)
F(−1, 3)
(1, 3)
D(−3, 1)
O
C(3, −1) x
B(−2, −1)
G(−1, −3)
E(1, −3)
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Example 12
A triangle ABC has vertices A(3, 1), B(7, 1), and C(7, 4).
a Plot the points A, B and C and draw the triangle ABC.
b Find coordinates of the vertices of the image of the triangle under a translation of 4 units
down followed by a clockwise rotation of 90° about the origin.
Solution
y
B(7, 1)
A(3, 1)
O
G
C′(7, 0) x
A″(−3, −3)
B′(7, −3)
N
AL
C″(0, −7)
PA
A′(3, −3)
B″(−3, −7)
ES
C(7, 4)
Exercise 18E
FI
You will need graph paper to do these exercises. Each question requires you to draw a
number plane with labelled x- and y-axes. A single number plane for each question is fine
(for example, one number plane for question 1 parts a–d), if points are neatly labelled. The
coordinates of the image point(s) should also be given.
Translations
Example
11a, b
1 Mark the following points on your graph paper and translate each one 2 units right and
3 units up. Be careful to label each point and its image. Also write down the coordinates of
each image point.
a A(0, 0)
b B(2, 1)
c C(−3, 1)
d D(−5, −5)
2 Given the following points and their images under a translation, plot the points and describe
the translation.
a A(0, 0) and A′ (2, 1)
460
b B(1, −2) and B′ (3, 3)
c C(0, 4 ) and C ′ (0, − 4 )
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1 8 E T r a n s f o r mat i o n s i n t h e C a r tes i a n p l a n e
Rotations
Example
11c, d
3 Mark the given points on a number plane and rotate them 90° anticlockwise about the
origin O. Write down the coordinates of each image point.
a A(2, −1)
b B(2, 1)
c C(−3, 1)
d D(0, 0)
4 Mark the given points on a number plane. Find their images under a rotation of 180°
anticlockwise about the origin O. Draw each image point and write down its coordinates.
a A(2, −1)
b B(2, 1)
c C(0, 0)
d D(0, 2)
Reflections
5 Copy the given points onto a number plane. Find their images when they are reflected in the
x-axis.
a A(2, −1)
b B(−3, 3)
ES
Example
11e, f
c C(−5, 0)
d D(0, 4)
6 Mark the given points on a number plane. Find their images under a reflection in the y-axis.
G
a i A(3, −1) ii B(−3, 3) iii C(0, 4)
PA
b Join the points A, B and C in part a to form Δ ABC. Find the image of Δ ABC under
reflection in the y-axis.
Combinations of transformations
Example 12
7 Mark the given points on a number plane, translate them 4 units to the right and rotate them
90° anticlockwise about the origin O. Write down the coordinates of each image point.
a A(2, −1)
b B(2, 1)
c C(−3, 1)
d D(−5, −5)
N
AL
8 Mark the given points on a number plane. Find their images when they are first reflected in
the x-axis and then in the y-axis.
a A(2, −1)
b B(2, 2)
c C(−3, 3)
d D(−5, 0)
9 Mark the given points on a number plane. Find the coordinates of their images when they
are first reflected in the x-axis and then translated 4 units down.
a A(0, 0)
b B(2, 1)
c C(−3, 1)
d D(−5, −5)
FI
10 A triangle ABC has vertices A(3, 1), B(7, 1), and C(7, 4).
a Plot the points A, B and C and draw the triangle ABC.
b Find the coordinates of the vertices of the image of the triangle under a translation of
4 units down followed by a reflection in the y-axis.
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461
18 F
Symmetry
Symmetry is an important idea in mathematics. We also see it in art and architecture. Here we are
going to consider two types of symmetry, namely rotational symmetry and reflection symmetry.
Reflection symmetry
ES
The figure shown opposite has reflection symmetry because it falls back
exactly on itself when it is reflected in the axis of reflection marked on the
diagram. This line is called an axis of symmetry of the figure.
Rotational symmetry
Think of rotating the figure shown opposite by 90° anticlockwise about O.
The figure moves back on top of the original figure where A has moved to
where B was, B has moved to where C was, C has moved to where D was,
and D has moved to where A was. We say that the figure has rotational
symmetry.
G
B
PA
Next we look at the situation after repeated anticlockwise rotations
of 90°. For example, the original point A in diagram 1 moves to the
point A in diagram 2. It then moves to the point A in diagram 3, and so on.
A
B
O
A
Diagram 1
D
O
N
AL
C
C
B
2 anticlockwise rotations of 90°
Diagram 3
C
B
O
FI
D
C
O
B
A
A
D
3 anticlockwise rotations of 90°
Diagram 4
You can see that it takes four rotations of
original position.
A
D
D
anticlockwise rotation of 90°
Diagram 2
O
C
4 anticlockwise rotations of 90°
Diagram 5
( )
360
4
°
= 90c° anticlockwise for the point A to return to its
We say that our figure has rotational symmetry of order 4:
• 90° is the smallest angle of rotation that makes it fit exactly on top of itself, and
• when we repeat this rotation four times, we get the identity rotation of 360°.
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1 8 F S y mmet r y
Can you see that this propeller
has rotational symmetry of order 3?
The smallest rotation that makes the propeller sit back on itself is 120°. When this is repeated
three times, we get the identity rotation of 360°.
Example 13
Solution
O
PA
G
a The four axes of symmetry are shown in
the solution diagram opposite. The axes of
symmetry are shown as dotted lines.
b The order of rotational symmetry is 4,
because the cross sits exactly on itself after
rotations of 90°, 180°, 270° and 360° about
the centre O.
ES
a Draw all axes of reflection symmetry
for the figure shown opposite.
b Write down the order of its rotational symmetry.
These examples show that figures may have more than one axis of symmetry, and figures may have
both rotational and reflection symmetry.
A figure can also have no rotational symmetry, for example, the number 5.
N
AL
Rotational symmetry
360°
is the smallest angle of rotation
n
that takes the figure so that it fits exactly on top of itself.
A figure has rotational symmetry of order n, n > 1, if
FI
Exercise 18F
Example 13
1 In the table on the next page, the letters of the alphabet and several common symbols are
drawn. Copy the table into your book.
a For each letter and symbol, mark any axes of symmetry.
b Does each figure have rotational symmetry? If so, mark the centre of the rotation on the
diagram. State the order of rotational symmetry.
Note: The symmetry of some letters may vary, depending on how you draw them.
Copy the letters accurately and comment on any issues of this type.
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463
1 8 F S y mmet r y
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
=
+
$
%
Order:
Order:
Order:
Order:
ES
Order:
2 In the following table, a number of common geometric figures are drawn. Copy the table
into your book.
a For each figure, mark any axes of reflection symmetry.
G
b Does each figure have rotational symmetry? If so, mark the centre of rotation on the
diagram and state the order of rotational symmetry.
N
AL
PA
Assume that the rectangle is not a square, the rhombus is not a square, the kite is not a
rhombus, the isosceles triangle is not equilateral, the trapezium has unequal vertical sides
and the parallelogram is not a rhombus.
square
rectangle
rhombus
kite
right triangle
trapezium
FI
Order:
equilateral
isosceles
triangle
triangle
circle
ellipse
Order:
parallelogram
Order:
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18 G
Regular polygons
A regular polygon is a plane figure in which every side is an interval
of the same length and all internal angles are equal. A polygon is
named after its number of angles; this is the same as the number of
sides. For example, the figure on the right is called a regular
pentagon, from the Greek words pente, meaning five, and gonos,
meaning angled. The equal angles are marked.
Number of sides
ES
The names of some common regular polygons are given in the
table below.
Name
Equilateral triangle
4
Square
5
Regular pentagon
6
Regular hexagon
7
Regular heptagon
8
Regular octagon
9
Regular nonagon
10
Regular decagon
12
Regular dodecagon
N
AL
PA
G
3
Drawing a regular pentagon
Use a protractor in the following to form angles of 72°.
B
B
C
A
FI
O
Draw a circle of
radius 5 cm. Mark
a radius OA.
C
72°
A
A
O
O
D
D
E
Draw rays 360° ÷ 5 = 72°
apart, starting at OA.
Let these rays intersect
the circle at B, C, D, E.
E
Join A, B, C, D, E to form
the regular pentagon.
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1 8 G Reg u l a r p o ly g o n s
Symmetry of regular polygons
3
4
Regular polygons have a lot of symmetry. For example, the regular pentagon
has five axes of reflection symmetry, as shown in the diagram opposite.
5
2
O
As we rotate the pentagon about the centre O, every 72° it will ‘fall
back on itself’, looking identical to the original diagram even though the
points have moved. The five rotations 72°, 144°, 216°, 288°, 360° about
the centre O are symmetries.
1
Thus the regular pentagon has rotational symmetry of order 5.
ES
Exercise 18G
1 a Draw a regular pentagon, using a circle of radius 6 cm and the method above.
b Mark the axes of reflection symmetry.
2 This question requires you to draw a regular nonagon (also called a 9-gon).
G
a Explain why the rays from the centre in your construction have to be 40° apart.
b Using a protractor, construct a regular nonagon in a circle of radius 5 cm.
PA
c Draw the axes of symmetry. Write down the order of rotational symmetry of a regular
nonagon.
3 a How many sides does a regular octagon have?
b Construct a regular octagon using only a straight edge and compasses by following the
steps below.
B
A
N
AL
A
D
C
FI
Construct a square of side
length 4 cm . Mark the
diagonals of the square to
locate its centre.
D
E
B
A
B
F
H
C
Place the point of your
compasses on the centre
of the square and draw
a circle around the square,
touching its four vertices.
Mark the midpoints of the
sides of the square.
C
D
G
Connect the opposite
midpoints with a line that
extends each end to meet
the circle. Join the points
A, E, B, F, C, G, D
and H to form a regular
octagon.
c What is the angle between adjacent rays from the centre in your construction?
d Draw the axes of symmetry and write down the order of rotational symmetry.
e Use a different colour to join every second vertex and so construct a regular 4-gon
(a square!). Which of the axes of symmetry of the octagon are axes of symmetry of the
square? What is the order of rotational symmetry of the square?
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1 8 G Reg u l a r p o ly g o n s
4 a Follow the steps below to construct a regular hexagon without using a protractor.
B
C
A
O
F
E
To check your accuracy,
draw a final arc centred
at F. It should pass
through A. As before,
join the points
A, B, C, D, E and F to
form a regular hexagon.
b Explain why this construction works.
G
ES
Place the point of your
compasses on A and
mark an arc intersecting
the circle at B. Move the
point of your compasses
to B and mark an arc
intersecting the circle
at C. Continue around
the circle.
A
O
D
F
E
Draw a circle of radius 5 cm.
Leave your compasses set at
5 cm and mark a point A on
the circle.
A
O
D
B
C
PA
c Draw the axes of reflection symmetry and write down the order of rotational symmetry.
5 a Construct a regular dodecagon in a circle of radius 5 cm, using the following steps.
A
B
A
G
B
L
C
E
D
Start by constructing a
hexagon using the method
in question 4. Mark the
midpoints of each side of the
hexagon.
H
C
F
I
K
N
AL
F
A
E
J
D
Connect the opposite
midpoints with a line
that extends each end to
meet the circle.
G
B
L
H
C
F
I
K
E
J
D
Join the points
A, G, B, H, C, I, D, J,
E, K, F and L to form a
regular dodecagon.
FI
b Label the vertices 1, 2, 3, …, 12 clockwise
(with 12 as the topmost vertex). What have you
constructed?
c How many axes of symmetry does this
polygon have, and what is its order of rotational
symmetry?
6 What are the orders of the rotational symmetry
of a regular 36-gon and a regular 37-gon?
Do not attempt to draw them! A regular 36-gon
is illustrated to the right, with straight lines
between adjected points.
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467
Review exercise
1 Describe the translation shown in each of these diagrams.
a
b
c
B
B
B′
A′
C
A
A
B′
A
A′
2 Rotate the interval AB by 60° in
a clockwise direction about
the point O.
C′
ES
A′
3 Rotate the triangle ABC by 90°
clockwise about O.
C
G
O
B
O
A
PA
A
B
N
AL
4 Rotate triangle ABC in the axis of
reflection.
5 Reflect triangle ABC in the axis of
reflection.
B
B
O
C
A
A
C
FI
6 Plot the points given below on graph paper and translate them 3 units right and 1 unit
up. Be careful to label each point and its image. Also write down the coordinates of each
image point.
a A(0, 0)
b B(2, 1)
c C(−3, 1)
7 Given a point and its image under translation in each part, plot the points and describe
each translation.
a A(0, 0) and A′ (3, 1)
b B(1, −3) and B′ (6, 2)
c C(0, 3) and C′ (0, −3)
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Re v i e w e x e r c i se
8 Plot the points given below on a number plane and rotate each one 90° anticlockwise
about the origin O. Write down the coordinates of each image point.
a A(1, −2)
b B(1, 4)
c C(−2, 1)
9 Plot these points on a number plane. Find their images when reflected in the x-axis.
a A(3, −2)
b B(3, 2)
c C(−2, 1)
b
c
d
PA
G
a
ES
10 State the order of rotational symmetry of each of the diagrams.
11 Perform each pair of successive translations on the given figure. Then find a single
translation that has the same effect.
b Translate 2 left, 3 up
then 1 left, 1 up.
N
AL
a Translate 1 right, 2 down
then 2 left, 1 down.
B
A
FI
C
B
A
C
12 Perform each pair of rotations on the given figure. Then find a single rotation that has
the same effect. All rotations are about O.
a Rotation of 30° anticlockwise then
60° anticlockwise.
B
b Rotation of 30° anticlockwise
then 60° clockwise.
C
B
A
O
C
A
O
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469
Re v i e w e x e r c i se
13 Plot the given points on the number plane. Reflect them in the x-axis and then reflect the
image in the y-axis. State the coordinates of the final image.
a A(5, −1)
b B(2, −1)
c C(−2, −3)
14 Plot the following points on your graph paper. First translate them 1 unit up and 3 units
to the right, and then translate the images 2 units up and 2 units to the left. State the
coordinates of the final images.
a A(1, −2)
b B(−5, −3)
c C(−1, 4)
15 In each part, describe two successive transformations that will have the combined effect
of mapping Δ ABC onto Δ A″B″C″.
b
A
B
A″
A
A″
ES
a
C
G
C
C″
B″
B
C″
B″
d
B
A
A″
PA
c
A
B
C″
C
N
AL
C
C″
B″
A″
FI
B″
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Challenge exercise
Answer these questions on square grid paper.
1 aReflect triangle ABC in line ℓ1 and then reflect the image in ℓ2. Label the first image
A′C′B′ and the second A″B″C″.
A
ℓ1
B
ES
C
ℓ2
b This time, reflect triangle ABC in line ℓ3 and then reflect the image in ℓ4. Label the
first image A′C′B′ and the second A″B″C″ as before.
A
B
PA
C
ℓ4
G
ℓ3
c Describe the translation that takes triangle ABC to triangle A″B″C″ in both parts a and
b. What do you notice?
N
AL
2 aReflect triangle ABC in line ℓ5 and then reflect the image in ℓ6. Label the first image
A′C′B′ and the second A″B″C″.
A
C
ℓ5
ℓ6
B
b Describe the translation that takes triangle ABC to triangle A″B″C″ in part a.
FI
c Describe the translation that takes triangle ABC to triangle A″B″C″ if the lines of
reflection are parallel 4 units apart.
d Suppose that the lines are a units apart. What is the translation?
3 The triangle ABC whose vertices have coordinates A(0, 0), B(2, 0), C(2, 4) is translated
to the triangle A′B′C′ with coordinates A′ (4, 0), B′ (6, 0), C′ (6, 4). Draw the two
triangles on a number plane and draw in two axes of reflection, ℓ1 and ℓ2, such that when
triangle ABC is reflected in ℓ1 and its image is reflected in ℓ2, the final result is triangle
A′B′C′. Is there only one way to do this? What is the distance between lines ℓ1 and ℓ2?
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471
C h a l l e n ge e x e r c i se
4 The triangle ABC whose vertices have coordinates A(0, 0), B(2, 0), C(2, 4) is
translated to the triangle A′B′C′ with coordinates A′ (0, 4), B′ (2, 4), C′ (2, 8). Draw the
two triangles on a number plane and draw in two axes of reflection, ℓ1 and ℓ2, such that
when triangle ABC is reflected in ℓ1 and its image is reflected in ℓ2, the final result is
triangle A′B′C′. Is there only one way to do this? What is the distance between lines ℓ1
and ℓ2?
a Draw two lines ℓ1 and ℓ2 through O such
that, when triangle ABC is reflected in ℓ1 and
its image is reflected in ℓ2, the final result is
triangle A′B′C′.
C′
A′
O
C
A
G
b Measure the acute angle between lines ℓ1 and ℓ2.
B′
ES
5 Triangle ABC has been rotated through 90°
in an anticlockwise direction about O to the
triangle A′B′C′. Copy the diagram carefully
into your book.
c Is there only one choice for the position of lines ℓ1 and ℓ2?
Circular graph paper (45°)
FI
N
AL
PA
Circular graph paper (30°)
Circular graph paper is available to download from your Interactive Textbook.
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B