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17 Transformations

17.1 Coordinates and translations

Worked example 1
axes axis
Translate rectangle ABCD 2 squares right and corresponds translate
4 squares up. Label it A’B’C’D’. Write down the
coordinates of the vertices A’, B’, C’ and D’.

y
5

0 x
–5 –4 –3 –2 –1 1 2 3 4 5
–1
A B
–2

–3
D C
–4

–5

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Continued

Move each vertex of the

y
5
rectangle 2 squares right and
4 squares up.
4
Vertex A on the original
A' B'
3 corresponds with vertex
2 Aˊ (you say A dash) on
the translated rectangle.
1 B corresponds with Bˊ, C
D' C'
corresponds with Cˊ and D
0 x corresponds with Dˊ.
–5 –4 –3 –2 –1 1 2 3 4 5
–1
A B Remember that when you
–2 write coordinates the x-axis
–3 number is first and the y-axis
D C
number is second.
–4

–5

Aˊ (-2, 3)
Bˊ (1, 3)
Cˊ (1, 1)
Dˊ (–2, 1)

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17.1 Coordinates and translations

Exercise 17.1
Focus
1 Match each point on the grid to its correct coordinates.
y
5

4
A
3

2
B
1

0 x
–5 –4 –3 –2 –1 1 2 3 4 5
–1

–2

–3
C
–4
D Tip
–5
(+, +) (right, up)
(+, -) (right, down)
i (2, –3) ii (–3, 1)
(-, +) (left, up)

iii (2, 3) iv (–3, –4) (-, -) (left, down)

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2 Here is an orienteering map. It shows the start and finish,

and checkpoints A, B, C and D on the route.
y

START/
3
FINISH
D
2

1
A

0 x
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6
–1

–2
B
–3

–4
C

Write down the coordinates of checkpoints A, B, C and D.

A B C D

3 The diagram shows four triangles.

y Tip
6
A Start with
5 triangle T and
B work out how
4
you can move
3 T to A, T to B,
T
2 and T to C.

1
C
0 x
1 2 3 4 5 6

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17.1 Coordinates and translations

Triangle T has been translated to triangles A, B and C.

Match each translation i, ii and iii to the correct triangle A, B and C.

i 1 square right and 2 squares down

ii 2 squares right and 3 squares up

iii 1 square left and 2 squares up.

Practice
4 The diagram shows line segments PQ and RS.
Write down the coordinates of the points P, Q, R and S.
y
5

2
Q
P
1

0 x
–5 –4 –3 –2 –1 1 2 3 4 5
–1
R
–2

–3
S
–4

–5

P Q R S

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5 Draw axes from –5 to +5 on squared paper. Draw a parallelogram

with vertices at E (–2, –2), F (2, –2), G (4, –4) and H (0, –4).

a Translate parallelogram EFGH 1 square right and 6 squares up.

Label the parallelogram EˊFˊGˊHˊ and write down the coordinates
of its vertices.

Eˊ Fˊ Gˊ Hˊ

b Translate parallelogram EFGH 3 squares left and 1 square down.

Label the parallelogram EˊˊFˊˊGˊˊHˊˊ and write down the coordinates
of its vertices.

Eˊˊ Fˊˊ Gˊˊ Hˊˊ

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17.1 Coordinates and translations

6 Draw axes from –6 to +6 on squared paper. Plot the points A (0, 1),
B (2, 1) and C (4, –2).

a Write down the coordinates of D so that A,

B, C and D are the vertices of an isosceles Tip
trapezium.
You can use
b Write down two possible coordinates of D fractions or
so that D is a point on the line segment AB. decimals in
coordinates.

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17 Transformations

c Write down two possible coordinates of D so that A, B, C and D

are the vertices of a parallelogram.

d Is it possible to find coordinates for D so that A, B, C and D are

the vertices of a rectangle? Explain your answer.

Challenge
7 (-1, 3) and (3, 1) are the coordinates of two vertices of a square.
What could the other vertices of the square be?
Find all the possible solutions.

8 The diagram shows shape P on a coordinate grid.

y
6

2
P
1

0 x
–5 –4 –3 –2 –1 1 2 3 4 5

Erin translates shape P 2 squares right and 3 squares up.

She labels the shape Q.

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17.1 Coordinates and translations

a What translation should Erin do to take shape Q back to shape P?

Explain how you worked out your answer.

b Erin translates shape Q 3 squares right and 1 square up.

She labels the shape R.
i Erin thinks that she could use the single translation 6 squares
right and 4 squares up to take shape P to shape R. Is Erin correct?
Explain your answer.

ii What do you notice about the single translation P to R, and the two
translations P to Q and Q to R?

9 Rectangle K has vertices as the points (-2, -1), (-5, -1), (-5, -3) and (-2, -3).
Shen translates K four times, using these four different translations
A, B, C and D.

A 3 squares right B 5 squares right

and 2 squares up and 6 squares up

C 3 squares right D 1 square left and

1 square down

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17 Transformations

After which translation will K and the new rectangle be:

a touching end to end

b touching corner to corner

c overlapping

d not touching or overlapping?

17.2 Reflections
Worked example 2
diagonal mirror line
Reflect this triangle in the diagonal mirror line.

Take one vertex of the triangle at a time.

Draw arrows (black) to the mirror line, then draw
the same length arrows (grey) the other side of the
mirror line. Join the vertices with straight lines to
complete the reflected triangle.

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17.2 Reflections

Exercise 17.2
Focus
1 Which drawings show correct reflections of triangle A?
a b

c d

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17 Transformations

2 Reflect each shape in the mirror lines. They have all been started for you.
a b

c d

3 Is A, B or C the correct reflection for each of these?

i

A B C

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17.2 Reflections

ii

A B C

iii

A B C

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Practice
4 Reflect the shape in the horizontal and vertical mirror lines.
a b

c d

5 This is part of Jose’s homework.

Question: Reflect shape A in the diagonal line of symmetry.
Label your answer shape B.

Has Jose drawn shape B correctly? Explain your answer.

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17.2 Reflections

6 Reflect the shape in the diagonal mirror lines.

a b

c d

7 a Describe the mirror line for each of these reflections.

i ii

iii

Tip

Is the mirror line

horizontal, vertical
or diagonal?

b Draw in the correct mirror line for each reflection.

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Challenge
8 Draw in the correct mirror line for each reflection.
a b c

9 a Reflect the shapes in the mirror lines to complete the pattern.

i ii

b What is the order of rotational symmetry of the completed pattern?

i ii

10 The diagram shows shape A on a coordinate grid. A

a Reflect shape A in the mirror line. Label the new shape B.
b Translate shape B 2 squares left and 1 square down.
Label the new shape C.
c Reflect shape C in the mirror line. Label the new shape D.
d Describe the translation that takes shape D back to shape A.

e What do you notice about your answer to part d and the translation
you carried out in part b?

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17.3 Rotations

17.3 Rotations
Worked example 3
anticlockwise centre of rotation
Rotate triangle A 90° anticlockwise clockwise corresponding rotate
about the centre of rotation marked C.
Label your answer triangle B.

Step 1 Step 2

A
A
C
C

Trace the shape, then put your point of Start turning the tracing paper 90°
your pencil on the centre of rotation. (a quarter turn) anticlockwise.

Step 3 Step 4

A
B C
C

Once the turn is completed make a note of Draw the new triangle onto the grid and
where the new triangle is. label it B.

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Exercise 17.3
Focus
1 In each of these diagrams, shape A has been rotated to shape B around
centre C. Write down if the rotation is clockwise or anticlockwise.
a A b Tip
A
C B Clockwise:
C
B

c d
B B
C A A
C Anticlockwise:

2 Complete these rotations of 90° clockwise

about the centre C.
a b C

3 Complete these rotations of 90° anticlockwise about the centre C.

a C b C

Practice
4 Rotate the shapes 90° clockwise about the centre C.
a b c C

C
C

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17.3 Rotations

5 Rotate the shapes 90° anticlockwise about the centre C.

a b c C
C

6 This is part of Alysha’s homework. The centre of rotation is shown by a dot (•).
Question:
Rotate shape A 90° clockwise about the centre of rotation (•).
Label the shape B.

B
Answer: A

Has Alysha got her homework correct?

Use diagrams to help you explain your answer.

Challenge
7 Rotate the shapes 90° about the centre of rotation C, using the direction shown.
a anticlockwise b clockwise

C C

8 a Follow these instructions to make a pattern.

1. Rotate the shape 90° clockwise about C.
C
2. Draw the new shape.
3. Rotate the new shape 90° clockwise about C.

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4. Draw the new shape.

5. Rotate the new shape 90° clockwise about C.
6. Draw the new shape.
b What is the order of rotational symmetry of your completed pattern?
9 a Rotate triangle A1B1C, using the same A1
instructions as question 8a.
Label the vertices of the three new C B1
triangles A2, B2, C then A3, B3, C then
A4, B4, C.
b On your completed diagram, join A1 to A2 to A3 to A4 to A1 with
straight lines. What shape have you just drawn?

c On your completed diagram, join B1 to B2 to B3 to B4 to B1 with

straight lines. What shape have you just drawn?

d Do you think that whatever shape you rotate, if you rotate it 90° clockwise
or anticlockwise three times, then the shape you get when you join
corresponding vertices will always be the same? Explain your answer.
You can use diagrams to help your explanation.

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