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14 Position and
transformation
14.1 Maps and plans
A scale drawing is a drawing that represents an object in real life.
Key words
The scale gives the relationship between the lengths on the drawing and
the real-life lengths. scale
An example of a scale is: scale drawing
1 cm represents 10 m.
So, 2 cm on the drawing represents 2 × 10 = 20 m in real life.
Tip
3 cm on the drawing represents 3 × 10 = 30 m in real life, etc.
This means
that 1 cm on
Exercise 14.1 the drawing
represents 10 m in
real life.
Focus
1 Complete these workings using a scale of 1 cm represents 5 m.
a 2 cm on the drawing represents 2 × 5 = m in real life. Tip
b 3 cm on the drawing represents 3 × 5 = m in real life. To go from the
c 8 cm on the drawing represents ×5= m in real life. drawing to real
2 Complete these workings using a scale of 1 cm represents 20 cm. life, you multiply
by the scale.
a 2 cm on the drawing represents 2 × 20 = cm in real life.
b 3 cm on the drawing represents × 20 = cm in real life.
c 6 cm on the drawing represents × = cm in real life. Tip
3 Complete these workings using a scale of 1 cm represents 10 m. To go from
real life to
a 20 m in real life represents 20 ÷ 10 = cm on the drawing.
the drawing,
b 30 m in real life represents 30 ÷ 10 = cm on the drawing. you divide by
c 70 m in real life represents ÷ 10 = cm on the drawing. the scale.

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14.1 Maps and plans

4 Complete these workings using a scale of 1 cm represents 50 cm.

a 100 cm in real life represents 100 ÷ 50 = cm on the drawing.
b 150 cm in real life represents ÷ 50 = cm on the drawing.
c 300 cm in real life represents ÷ = cm on the drawing.
5 Look at the cards shown. The white cards are scale drawing
measurements. The grey cards are real-life measurements.
Match each white card (A to G) to the correct grey card (i to vii).
Scale: 1 cm represents 2 m.
The first one has been done for you: A and iv.
F
2 cm 6 cm 5 cm 2.5 cm 12 cm 3.5 cm 9.5 cm

iv v vi vii
10 m 12 m 7m 4m 19 m 5m 24 m

Practice
6 Zara makes a scale drawing of the front of a large building.
She uses a scale of 1 cm represents 5 m.
a On her drawing the building is 16 cm long. How long is the
building in real life?
b The building in real life is 120 m tall. How tall is the building
on the scale drawing?
7 Sofia makes a scale drawing of a window. She uses a scale of 1 cm
represents 20 cm.
a On her drawing the window is 6 cm wide. How wide is the
window in real life? Give your answer in metres.
b The window in real life is 2.2 m tall. How tall is the window on
the scale drawing? Give your answer in centimetres.
8 The map shows part of Spain.
The scale of the map is 1 cm represents
30 km.
a Use a ruler to measure the distance, in
cm, from Madrid to Toledo.
b Work out the distance, in km, from
Madrid to Toledo in real life.

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14 Position and transformation

9 Arun makes a scale drawing of his garden. He uses

centimetre-squared paper. He uses a scale of 1 to 200.
a Calculate the distance in real life of each of A B
these lengths. Give your answers in metres.
i AB ii BC
iii CD iv DE
D C
v EF vi AF
b The path in Arun’s garden is 11 m long. How
long will the path be on the scale drawing?
Give your answer in centimetres.
c The flowerbed in Arun’s garden is 3 m wide. F E
How wide will the flowerbed be on the scale
drawing? Give your answer in centimetres.
10 A map has a scale of 1 : 12 000.
a On the map the distance between two shops is 20 cm. What is
the distance, in km, between the two shops in real life?
b The distance between two buildings is 6 km in real life. What is
the distance, in cm, between the two buildings on the map?

Challenge
11 This map has a scale of 1 : 90 000.
a Use a ruler to measure the distance,
in cm, from St. Patrick to Six
Cross Roads.
b Work out the distance, in km, from
St. Patrick to Six Cross Roads in
real life.
c The distance from St. Patrick to
Pothouse is 10.35 km in real life.
Work out how far this is, in cm, on
the map. Tip
12 Rihanna takes part in a triathlon. She is given a map of the route A triathlon is a
for each part of the triathlon. Each map has a different scale. competition in
Rihanna measures the distance of the route on each map. The table three parts. In
shows the distances on each map and each map’s scale. the first part you
must swim, in the
second part you
must cycle and in
the third part you
must run.

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14.1 Maps and plans

Part of Distance on Distance in

Map scale
triathlon map (cm) reallife (km)
swim 1 : 15 000 18
cycle 1 : 400 000 16
run 1 : 80 000 20.5
a Copy and complete the table.
b What is the total distance, in km, of the triathlon?
13 Sven takes part in a 56 km cycle ride. The distance of the route on a
map is 32 cm.
Work out if A, B or C is the correct map scale. Show your working.
A 1 : 125 000 B 1 : 150 000
000 C 1 : 175 000
14 The distance from town A to town B is 18 cm on the map, and is
27 km in real life. The distance from town B to town C is 35 cm on
the map.
How far is the distance from town B to town C in real life? Show
how you worked out your answer.

15 A police officer uses the length of a footprint to estimate the height of a criminal. The scale
they use is 2 : 13.
Estimate the height, in metres, of the criminals who have left these footprints. Show all
your working.
a b c

236 mm
26 cm
30 cm

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14 Position and transformation

14.2 Distance between two points

Azra uses this method to work out the distance between the two points
(8, 4) and (3, 4). Key words
Step 1 Write the coordinates underneath each other. (8, 4 ) distance between
(3, 4 ) two points
Step 2 Put your finger over the numbers that are the same. (8, 4 ) x-coordinate
(3, 4 )
Step 3 There are two numbers left, so work out
biggest number − smallest number.
8 − 3 = 5 units

Exercise 14.2
Focus
1 Work out the distance between these pairs of points. Use Azra’s
method, which is shown in the introduction. The first one has been
started for you.
a (9, 2) and (6, 2) (9, 2 )
(6, 2 )
b (12, 7) and (2, 7) c (1, 0) and (8, 0) d (12, 11) and (20, 11)
2 Work out the distance between these pairs of points. Use Azra’s
method, which is shown in the introduction. The first one has been
started for you.
a (1, 5) and (1, 9) (1 , 5) 9−5= units
(1 , 9)
b (2, 8) and (2, 3) c (9, 15) and (9, 10) d (11, 18) and (11, 28)

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14.2 Distance between two points

3 Look at the cards shown. The white cards show pairs of points.
The grey cards show the distances between the points.
Match each white card (A to G) to the correct grey card
(i to vii). The first one has been done for you: A and iii.
A B C D E F G
(1, 8) (8, 10) (1, 5) (13, 7) (15, 15) (18, 2) (0, 27)
(1, 3) (8, 12) (7, 5) (9, 7) (15, 14) (21, 2) (0, 20)

i ii iii iv v vi vii
2 units 4 units 5 units 7 units 3 units 1 unit 6 units

Practice
4 Work out the distance between these pairs of points. Each pair of
points has the same y-coordinate.
a (14, 2) and (18, 2) b (10, 8) and (5, 8) c (2, 0) and (10, 0)
5 Work out the distance between these pairs of points. Each pair of
points has the same x-coordinate.
a (3, 17) and (3, 5) b (18, 15) and (18, 25) c (9, 19) and (9, 21)
6 Work out the distance between each pair of points. Choose the correct answer: A, B or C.
a (9, 14) and (9, 23)
A 5 units B 7 units C 9 units
b (16, 4) and (0, 4)
A 12 units B 16 units C 0 units
c (3, 17) and (3, 16)
A 1 unit B 13 units C 14 units
d (8, 8) and (19, 8)
A 12 units B 0 units C 11 units

7 This selection of cards show the coordinates of the points A to J.

A (14, 9) B (17, 18) C (21, 9) D
D (17, 27)
E (14, 16) F (26, 18) G
G (14, 2) H (8, 18)
I (7, 9) J (17, 9)
Classify the cards into these two groups:
Group 1: Points that are 7 units from A.
Group 2: Points that are 9 units from B.

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14 Position and transformation

Challenge
8 Quadrilateral ABCD has vertices at the points A(4, 7), B(9, 7), C(9, 11)
and D(4, 11).
Arun and Zara are discussing the quadrilateral.
Arun says: Zara says:

I think ABCD is I think ABCD is

a square. a rectangle.

Who is correct? Show your working and explain your answer.

9 The grey cards show pairs of points on a coordinate grid.
The white cards show the distances between two points.
Match each grey card (A to D) to its correct white card (i to iv).
A −9, 5) and (−4, 5) B (8, −3) and (8, −9)
C −5, −4) and (−5, −12) D −3, 0) and (−10, 0)
i 6 ii 7 iii 5 iv 8
10 Work out the distance between each pair of points.
a (7, 4) and (7, −4) b (−3, 2) and (6, 2)
c (−8, 10) and (−8, −2) d (20, −1) and (−12, −1)
11 Quadrilateral EFGH has vertices at the points E(−4, 1), F(−1, 3), G(2, 1)
and H(−1, −5).
a Work out the lengths of the diagonals:
i EG ii FH
b Use your answers to part a to explain why:
i EFGH is not a square ii EFGH is not a rectangle
c The diagonals EG and FH cross at the point X. Mair thinks that X
has coordinates (−1, 1). Explain how you can tell, by looking at the
coordinates of E, F, G and H, that Mair is correct.
d Work out the lengths of:
i EX ii XG iii FX iv XH
e Use your answers to part d to explain why:
i EFGH is not a rhombus ii EFGH is a kite

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14.3 Translating 2D shapes

12 P is the point (2, 3). The distance PQ is 9 units. Work out four
possible coordinates for the point Q. Show your working.

14.3 Translating 2D shapes

The diagram shows triangles ABC and A′B′C′.
Key words
A′B′C′ is the image of ABC.
The vertices of ABC have coordinates A(3, 3), B(3, 1) and C(4, 1). image
ABC is translated 2 squares right, to become A′B′C′. object
The vertices of A′B′C′ have coordinates A′(5, 3), B′(5, 1) and C′(6, 1).
You add 2 to the x-coordinates of ABC to get the coordinates of A′B′C′.
A (3, 3) B (3, 1) C (4, 1)
4
 +2  +2  +2 3
A′ (5, 3) B′ (5, 1) C′ (6, 1) 2
1
The diagram shows triangles DEF and D′E′F′.
D′E′F′ is the image of DEF. 0
The vertices of DEF have coordinates D(2, 5), E(2, 3) and F(3, 3).
DEF is translated 3 squares up, to become D′E′F′.
8
The vertices of D ′E′F′ have coordinates D′(2, 8), E′(2, 6) and F′(3, 6). 7
You add 3 to the y-coordinates of DEF to get the coordinates 6
of D′E′F′. 5
D (2, 5) E (2, 3) F (3, 3) 4
3 F
 +3  +3  +3 2
1
D′ (2, 8) E′ (2, 6) F′ (3, 6)
0 1 2

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14 Position and transformation

Exercise 14.3
Focus
1 Triangle ABC from the first example in the introduction, is translated
5 squares right.
Copy and complete the workings to find the coordinates of the
vertices of the image A′B′C′.
A (3, 3) B (3, 1) C (4, 1)
 +5  +5  +5
A′ (8, 3) B′ ( , 1) C′ ( , 1)
2 Triangle ABC from the first example in the introduction, is
translated 3 squares left. Tip
Copy and complete the workings to find the coordinates of the When you
vertices of the image A′B′C′. translate a shape
3 squares left, you
A (3, 3) B (3, 1) C (4, 1)
subtract 3 from
 −3  −3  −3 the x-coordinates
of ABC to get
A′ (0, 3) B′ ( , 1) C′ ( , 1) the coordinates
3 A square, ABCD, has vertices at the points A(5, 3), B(8, 3), of A’B’C’.
C(8, 6) and D(5, 6).
ABCD is translated 4 squares left. The image of ABCD is A′B′C′D′.
Copy and complete the workings to find the coordinates of the vertices
of A′B′C′D′.
A (5, 3) B (8, 3) C (8, 6) D (5, 6)
 −4  −4  −4  −4
A′ (1, 3) B′ ( , 3) C′ ( , 6) D′ ( , 6)
4 Triangle DEF from the second example in the introduction, is translated
1 square up.
Copy and complete the workings to find the coordinates of the vertices of
the image D′E′F′.
D (2, 5) E (2, 3) F (3, 3)
 +1  +1  +1
D′ (2, 6) E′ (2, ) F′ (3, )

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14.3 Translating 2D shapes

5 Triangle DEF from the second example in the introduction, is Tip

translated 2 squares down.
Copy and complete the workings to find the coordinates of the When you
vertices of the image D′E′F′. translate a shape
2 squares down,
D (2, 5) E (2, 3) F (3, 3) you subtract 2
 −2  −2  −2 from the
y-coordinates
D′ (2, 3) E′ (2, ) F′ (3, ) of DEF to get
6 A rectangle, ABCD, has vertices at the points A(2, 7), B(6, 7), the coordinates
C(6, 9) and D(2, 9). of D’E’F’.
ABCD is translated 4 squares down. The image of ABCD
is A′B′C′D′.
Copy and complete the workings to find the coordinates of the
vertices of A′B′C′D′.
A (2, 7) B (6, 7) C (6, 9) D (2, 9)
 −4  −4  −4  −4
A′ (2, 3) B′ (6, ) C′ (6, ) D′ (2, )

Practice
7 A triangle, PQR, has vertices at the points P(4, 5), Q(7, 5) and R(6, 9).
PQR is translated 3 squares right and 4 squares up. The image of PQR is P′Q′R′.
Copy and complete the workings to find the coordinates of the vertices
of P′Q′R′.
P (2, 5) Q (7, 1) R (2, 5)
 +3 +4  +3 +4  +3 +4
P′ (7, ) Q′ ( , 9) R′ ( , )
8 A rectangle, ABCD, has vertices at the points A(4, 8), B(8, 8), C(8, 10) and
D(4, 10).
ABCD is translated 1 square left and 5 squares down. The image of ABCD
is A′B′C′D′.
Copy and complete the workings to find the coordinates of the vertices
of A′B′C′D′.
A (4, 8) B (8, 8) C (8, 10) D (4, 10)
 −1 −5  −1 −5  −1 −5  −1 −5
A′ (3, ) B′ ( , 3) C′ ( , ) D′ ( , )

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14 Position and transformation

9 The grey cards have different translations written on them. The

white cards show what must be added or subtracted to the x- and
y-coordinates of a shape to complete the translation.
Match each grey card (A to D) to its correct white card (i to iv).
The first one has been done for you: A and iv.
A 6 squares right and 3 squares down
B 6 squares left and 3 squares up
C 6 squares left and 3 squares down
D 6 squares right and 3 squares up

−6−3 +6  + 3

−6+3 +6  − 3

10 A rhombus, PQRS, has vertices at the points P(6, 2), Q(7, 4),
R(6, 6) and S(5, 4).
PQRS is translated 3 squares left and 2 squares down. The image
of PQRS is P′Q′R′S′.
Work out the coordinates of the vertices of P′Q′R′S′.
11 This is part of Ivan’s homework.

Ivan has worked out the coordinates of A′ and B′ incorrectly.

a Explain the mistake that he has made.
b Work out the correct coordinates of A′ and B′.
c Explain how Ivan could check that his answers are correct.

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14.3 Translating 2D shapes

12 Simone translates the triangle XYZ to X′Y′Z′.

XYZ has vertices at X(5, 4), Y(8, 5) and Z(6, 9).
X′Y′Z′ has vertices at X′(2, 7), Y′(5, 8) and Z′(3, 12).
Which translation, A, B, C or D, has she used? Show your working
and explain your answer.
A 3 squares right and 3 squares down
B 3 squares right and 3 squares up
C 3 squares left and 3 squares down
D 3 squares left and 3 squares up

Challenge
13 Alya translates kite JKLM to J′K′L′M′. JKLM has vertices at
J(1, 5), K(4, 4), L(1, −1) and M(−2, 4).
Alya works out that the vertices of J′K′L′M′ are at J′(−2, 4),
K′(1, 3), L′(−2, 2) and M′(−5, 3).
Alya has worked out three of the vertices correctly and
one incorrectly.
Which vertex, J′, K′, L′ or M′, is incorrect? Explain how you
worked out your answer.
14 A rectangle, CDEF, has vertices at the points C(2, −1), D(6, −1),
E(6, −3) and F(2, −3).
Nadim translates CDEF four times, using four different translations.
He writes down the coordinates of the images of CDEF after each
translation. These are the four translations he uses:
A 5 squares right and 3 squares up
B 4 squares right and 2 squares down
C 4 squares left
D 3 squares left and 1 square down
After which translation will the object and the image be:
a touching end to end? Tip
b touching corner to corner? Remember that
c overlapping? you must not
d not touching or overlapping? draw a grid to
help you answer
Explain how you worked out your answers. this question.

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14 Position and transformation

15 a Jon translates shape L 3 squares right and 2 squares up.

He labels the shape M.
Jon then translates shape M 4 squares right and 6 squares up.
He labels the shape N.
i What translation could Jon do on shape L to take it to Tip
shape N in one step?
You could
ii Explain the method you used to work out your answer to choose your
part a i. own coordinates
iii What do you notice about your answer to part a i and for one of the
Jon’s two translations? vertices of shape
b Jim translates shape L 2 squares left and 4 squares up. L and test the
He labels the shape P. translations on
Jim then translates shape P 3 squares left and 1 square up. this point.
He labels the shape Q.
i What translation could Jim do on shape L to take it to
shape Q in one step?
ii Explain the method you used to work out your answer to
part b i.
iii What do you notice about your answer to part b i and
Jim’s two translations?
16 In a game of chess, in one turn, a knight can move either
1 square left or right followed by 2 squares up or down or 2
squares left or right followed by 1 square up or down. On the
chess board shown, K shows the position of the knight.
a i Describe the translations that the knight must
make to get to the square marked X.
ii What is the fewest number of moves that the
knight can make to get from K to X?
b i Describe the translations that the knight must
make to get to the square marked Y.
ii What is the fewest number of moves that the knight
can make to get from K to Y?

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14.4 Reflecting shapes

14.4 Reflecting shapes

Exercise 14.4 Key word
reflected
Focus
1 In each of these diagrams, the shape is reflected in the x-axis. Copy and complete each diagram.
a b c
4 4 4
3 3 3
2 2 2
1 1 1
0 0
–1 –1 –1
–2 –2 –2
–3 –3 –3
–4 –4 –4

2 In which of these diagrams has shape A been correctly reflected in the x-axis?
If the reflection is incorrect, copy the diagram and draw the correct reflection.
a b c
4 4 4
3 3 3
2 2 2
1 1 1
0 0 0
–1 –1 –1
–2 –2 –2
–3 –3 –3
–4 –4 –4

3 In each of these diagrams, the shape is reflected in the y-axis. Copy and complete
each diagram.
a b c
4 4 4
3 3 3
2 2 2
1 1 1

–4 –3 –2 –10 1 2 3 4 x –4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x

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14 Position and transformation

4 In which of these diagrams has shape B been correctly reflected in the y-axis?
If the reflection is incorrect, copy the diagram and draw the correct reflection.
a b c
4 4 4
3
2 2 2
1 1 1
–4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x

Practice
5 Copy each diagram and reflect the shape in the x-axis.
a b c
4 4 4
3 3 3
2 2 2
1 1 1
0 0
–1 –1 –1
–2 –2 –2
–3 –3 –3
–4 –4 –4

6 Copy each diagram and reflect the shape in the y-axis.

a 4 b 4 c
4
3 3 3
2 2 2
1 1 1
–4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x
7 Copy each diagram and reflect each letter shape in:
i the x-axis ii the y-axis
a b
4 4
3 3
2 2
1 1
–4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x
–1 –1
–2 –2
–3 –3
–4 –4

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14.4 Reflecting shapes

c d
4 4
3 3
2 2
1 1
–4 –3 –2 –10 1 2 3 4x –4 –3 –2 –10 1 2 3 4x
–1 –1
–2 –2
–3 –3
–4 –4

8 The diagram shows ten trapezia, which are labelled A to J.

Tip
Copy and complete these statements. The first one has been done
for you. ‘Trapezia’ is
a A is a reflection of G in the x-axis.
5 the plural
of ‘trapezium’.
b D is a reflection of in the ……. 3
2
c E is a reflection of in the ……. 1
d J is a reflection of in the ……. –5 –4 –3 –2 –10
–1
F
e H is a reflection of in the ……. –2

f F is a reflection of in the ……. G

–4
–5
Challenge
9 a Helena reflects rectangle JKLM in the x-axis. The vertices of the
rectangle have the coordinates J(2, 1), K(5, 1), L(5, 3) and M(2, 3).
The table shows the coordinates of the vertices of the object and
the vertices of its image.
Copy and complete the table.
Explain the method you used to work out your answers.

Object J(2, 1) K(5, 1) L(5, 3) M(2, 3)

Image J′( , ) K′( , ) L′( , ) M′( , )

b Helena reflects rectangle JKLM in the y-axis.

The table shows the coordinates of the vertices of the object and
the vertices of its image.
Copy and complete the table.
Explain the method you used to work out your answers.

Object J(2, 1) K(5, 1) L(5, 3) M(2, 3)

Image J′( , ) K′( , ) L′( , ) M′( , )

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14 Position and transformation

10 Make a copy of this diagram.

5
a Reflect shape A in the x-axis. Label the shape B. 4
b Reflect shape A in the y-axis. Label the shape C. 3
2
c Reflect shape B in the y-axis. Label the shape D. 1
d Colour in the combined shape of A, B, C and D. –5 –4 –3 –2 –10 1 2 3 4 5x
–1
i How many lines of symmetry does your combined –2
shape have? –3
ii What is the order of rotation of your combined shape? –4
–5

11 Copy each diagram and reflect the shape in the y-axis.

a b c
4 4 4
3 3 3
2 2 2
1 1 1
–3 –2 –10 1 2 3x –3 –2 –10 1 2 3x –3 –2 –10 1 2 3x

12 The diagram shows kite PQRS.

6
The vertices of the kite have coordinates 5
P(1, 3), Q(4, 1), R(1, −5) and S(−2, 1). 4
3
Arun is going to reflect PQRS in the x-axis. 2
1
–5 –4 –3 –2 –10
–1
I’m going to work out the –2
coordinates of P’, Q’, R’ and S’ –3
first and then draw P’Q’R’S’. The –4
coordinates are: P’(1, −3), –5
Q’(4, −1), R’(−1, −5) and S’(2, 1). –6

Arun has made some mistakes with his coordinates for P′Q′R′S′.
a Without drawing P′Q′R′S′, work out which coordinates are
incorrect. Explain the mistakes Arun has made.
b Copy the diagram. Work out the correct coordinates for
P′Q′R′S′ and draw P′Q′R′S′ on the diagram.

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14.5 Rotating shapes

14.5 Rotating shapes

When you rotate a shape, you turn it about a fixed point called the centre
of rotation. Key words
The centre of rotation is usually shown as a dot • with the letter C. anticlockwise
You turn a shape clockwise or anticlockwise . centre of rotation
You must give the number of degrees by which you are rotating the object. clockwise
The rotations that are most often used are 90° and 180°.

Tip
Remember that
Exercise 14.5 when a shape (the
object) is rotated
to a new position
Focus (the image), the
1 For each of the following, write the number of degrees shape A has object and the
been turned to get to shape B. image are always
Describe if the turn is clockwise or anticlockwise. The first one has congruent.
been done for you.
a b c
A
B
B C

90° clockwise

d e f
B
B

g h

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14 Position and transformation

2 Copy each diagram. Rotate each shape 90° clockwise about the centre of rotation, C.
a b c
C

3 Copy each diagram. Rotate each shape 180° about the centre of rotation, C.
a b c
C
C

Practice
4 Copy each diagram and rotate the shape about the centre, C, by the given number of degrees.
a b c d

180° 90° anticlockwise 90° clockwise 180°

5 Copy each diagram and rotate the shape about the centre, C, by the given number of degrees.
a b c d

180° 90° clockwise 180° 90° anticlockwise

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14.5 Rotating shapes

6 Copy each diagram and rotate the shape, using the

information given.

a b
6 6

4 4

2 2

0 0
0 1 2 3 4 5 0 1 2 3 4 5

90° anticlockwise 180°

centre (2, 4) centre (2, 3)

c d
6 6

4 4

2 2

0 0
0 1 2 3 4 5 0 1 2 3 4 5

90° clockwise 180°

centre (2, 2) centre (3, 3)
7 a Copy this diagram.
Follow these instructions to make a pattern.
i Rotate the pattern 90° anticlockwise about the centre, C.
ii Draw the image.
iii Rotate this new image 90° anticlockwise about the
centre, C.
iv Draw the image.
v Rotate the third image 90° anticlockwise about the
centre, C.
vi Draw the image.
b What is the order of rotational symmetry of the
completed pattern?
c How many lines of symmetry does the completed pattern have?

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14 Position and transformation

Challenge
8 Zara draws rectangle A onto this coordinate grid. Each square on
the grid has a length of 1 cm. 4
a What is the perimeter of rectangle A? 3
Zara rotates rectangle A 180° about the centre (4, 2) to get 2
rectangle B. She joins rectangle A to rectangle B to give a 1
combined shape. 0
Zara says:

Tip
The perimeter of my You could use
combined shape is a diagram to
twice the perimeter of help you prove
rectangle A. or disprove
Zara’s comment.
b Is Zara correct? Give reasons for your answer.
9 Make two copies of the diagram shown in Question 8.
a Using the first copy, rotate the rectangle 90° clockwise about the
centre (4, 1). The object and image together make one shape.
i What is the perimeter of this shape?
ii What do you notice about the perimeter of the rectangle
and the perimeter of the shape?
b Using the second copy, rotate the rectangle 180° about the
centre (4, 3). The object and image together make one shape.
i What is the perimeter of this shape?
ii What do you notice about the perimeter of the rectangle
and the perimeter of the shape?
10 a Copy this diagram.
4
b Rotate rectangle P 180° about the centre (4, 3). Label the 3
image Q. 2
Shade in the combined shape of rectangles P and Q. 1

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14.6 Enlarging shapes

c Sofia says:

5
Are Sofia’s statements correct? Explain your answers. 4
3
11 Make two copies of this diagram.
2
a Using the first copy, rotate the shape 90° clockwise 1
about the centre (3, 1).
0
b On the second copy, rotate the shape 180° about the –1
centre (4, 2).

14.6 Enlarging shapes Key words

dimensions
An enlargement of a shape is a copy of the shape that is bigger than the enlargement
original shape. scale factor
The original shape and the enlargement are similar shapes. similar shapes
You enlarge a shape using a scale factor.
When the scale factor is 2, all the sides of the image must be twice as
long as the object. Tips
When the scale factor is 3, all the sides of the image must be three times ‘twice as long’
as long as the object. means two times
the length.
‘three times as
long’ means three
times the length.

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14 Position and transformation

Exercise 14.6
Focus
1 Copy and complete these enlargements. Use a scale factor of 2.
a b

c d

2 Copy and complete these enlargements. Use a scale factor of 3.

a b

c d

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14.6 Enlarging shapes

3 Copy each of these shapes onto squared paper. Enlarge each shape using the scale
factors given.
a b c

scale factor 2 scale factor 3 scale factor 4

Practice
4 Copy each of these shapes onto squared paper. Enlarge each shape using the scale
factors given.
a b c

scale factor 2 scale factor 3 scale factor 4

5 This is part of Dan’s homework.

a Dan has made some mistakes. Explain the mistakes he has made.
b Draw the correct solution.

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14 Position and transformation

6 The diagram shows a shape and its enlargement.

object image

What is the scale factor of the enlargement?

7 Here are some triangles. The triangles are not drawn accurately.
A B C

10 cm
4 cm 6 cm
2 cm

3 cm 5 cm 9 cm 16 cm
a Explain why triangle B is not an enlargement of triangle A.
b Are triangles C and D an enlargement of triangle A? Explain how
you worked out your answers.

Challenge
8 A company makes boxes with square bases. The diagram shows 5 cm
the dimensions of the smallest box that it makes.
These are the dimensions of the other boxes that the company 4 cm
makes. Some of the boxes are enlargements of the smallest box.
Some of the boxes are not enlargements of the smallest box.

10 cm 40 cm
30 cm
8 cm 15 cm 25 cm
B 24 cm 32 cm
10 cm 16 cm

Which of these boxes are enlargements of the smallest box?

Write down the scale factor of each enlargement.

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14.6 Enlarging shapes

9 Marcus and Arun look at this diagram.

Marcus says: Arun says:

I think that shape I don’t think

B is an enlargement of that shape A is an
shape A because all the enlargement of shape B
sides are twice because the shape has
as long. been rotated 90°.

Who do you think is correct, Marcus or Arun? Explain

your answer.
10 Naveen enlarges triangle A to get triangle B. He then enlarges
triangle B to get triangle C. Finally, he enlarges triangle C to get
triangle D.
triangle A triangle B triangle C triangle D

height 96 cm
height
4 cm

3 cm base base 72 cm
a i Work out one possible base length and height for
triangles B and C.
ii Explain how you worked out your answers.
iii Give the scale factor you used for each enlargement.
b i Work out a different possible base length and height for
triangles B and C.
ii Explain how you worked out your answers.
iii Give the scale factor you used for each enlargement.
c i How many different base lengths and heights could there
be for triangles B and C?
ii Explain how you worked out your answer.

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14 Position and transformation

11 The diagram shows four rectangles, A, B, C and D.

a Write the scale factor of the enlargement of rectangle:

i A to B ii A to C iii A to D
b Work out the area of rectangle:
i A ii B iii C iv D
c Copy and complete this table. Write each ratio in its
simplest form.

Scale factor of Ratio Ratio

Rectangles
enlargement of lengths of areas
A:B 2 1:2
A:C
A:D
Tip
d Write a rule that connects the ratio of lengths to the ratio Remember the
of areas. square numbers:
e Will this rule work for any scale factor of enlargement?
12 = 1, 22 = 4,
Will this rule work, in general, for any shape? 32 = 9, 42 = 16, etc.
Explain your answers.

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