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Transformations

Worksheets for GCSEMathematics

Maths Resources for Teachers

Shape

Transformations Worksheets

Contents

Differentiated Independent Learning Worksheets

Solutions

• Reflective Symmetry• Rotational Symmetry• Enlarging Shapes• Rotating Shapes• Translating Shapes• Reflections on a Grid• Rotations on a Grid• Translation Vectors• Enlargements on a Grid• Negative Scale Factor Enlargements• Describing Transformations• Vector Addition• Vector Geometry

Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15

• Reflective Symmetry• Rotational Symmetry• Enlarging Shapes• Rotating Shapes• Translating Shapes• Reflections on a Grid• Rotations on a Grid• Translation Vectors• Enlargements on a Grid• Negative scale factor enlargements• Describing Transformations• Vector Addition• Vector Geometry

Page 16Page 18Page 19Page 20Page 22Page 23Page 24Page 25Page 26Page 27Page 28Page 29Page 30

2

Reflective Symmetry

Q1. Draw all the lines of symmetry for each shape.

a) b) c) d)

e) f) g) h)

Q2. Arrange these shapes in order of least to most lines of reflective symmetry.

Q3. Copy these diagrams and reflect them in the dotted lines.

a) b) c) d)

Q4. Copy these diagrams and reflect them in the dotted lines. a) b) c)

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Rotational Symmetry

Q1. Which of these shapes have rotational symmetry? State the order of each shape with rotational symmetry.

a) b) c) d)

e) f) g) h)

Q2. Which of these shapes is the odd one out? Why?

Q3. Copy and complete these diagrams so that they have the rotational symmetry of order 4 with center

at .

a) b) c) d)

Q4. Complete the following diagrams to give the order or rotational symmetry stated.

a) Shade 4 squares for order 4 b) Shade 3 squares for order 2 c) Shade 3 squares for order 4

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Enlarging Shapes

Q1. Enlarge each of these shapes to the scale factor given. a) b) c)

Scale Factor 2 Scale Factor 2 Scale Factor 3 d) e) f)

Scale Factor 3 Scale Factor ⁄ Scale Factor ⁄

Q2. Enlarge each of these shapes to the scale factor and center stated. a) b) c)

Scale Factor 2 Scale Factor 3 Scale Factor 2 d) e) f)

Scale Factor 2.5 Scale Factor ⁄ Scale Factor ⁄

Q3. Describe the enlargement including the center and scale factor that maps the smaller triangle onto the larger one.

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Rotating Objects

Q1. Rotate the following images a quarter turn anti-clockwise about the center.

a) b) c)

d) e) f)

Q2. Rotate the following images to the direction and center shown.

a)

i) ⁄ turn CW about A

ii) ⁄ turn about B

b)

i) ⁄ turn ACW about C

ii) ⁄ turn about D

c)

i) ⁄ turn about E

ii) ⁄ turn ACW about F

Q3. Rotate the following images by 90°, 180° and 270° clockwise about the center.

a) b) c)

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Translating Objects

Q1. Describe the following translations using words.

Q2. Copy these shapes on squared paper and translate using the translations given.

a) b) c) d)

3 left, 2 down 5 Left 2 right, 5 up 3 down

e) f) g) h)

4 right, 1 down 5 left, 2 down 4 right, 3 down 1 left, 4 up

Q3. Simon starts in square x facing north. He moves one square forward, turns to the right, moves one square forward, turns to the left and moves three squares forward again.

Describe the translation from his start and end positions.

Q4. I translate a shape 5 right and 6 up, then 8 left and 10 down. Write down a single transformation that would do the same single translation.

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Reflections on a Grid

Q1

a)

b)

c)

d)

Make a copy of the diagram and complete the following reflections.

Reflect triangle A in the line . Label the new triangle A’.

Reflect triangle A in the line . Label the new triangle B’.

Reflect triangle B in the line Label the new triangle B’.

Reflect triangle B in the line Label the new triangle B’’.

Q2. Make a copy of the diagram and complete the following reflections.

Reflect triangle H in the line . Label the new triangle I.

Reflect triangle H in the line . Label the new triangle J.

Reflect triangle H in the line Label the new triangle K.

Q3.

a)

b)

c)

Describe fully the reflective transformation that maps:

Shape W onto X

Shape W onto Y

Shape W onto Z

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Rotations on a Grid

Q1.

a)

b)

c)

d)

Copy the diagram to the right and rotate the given triangle by the following:

90° clockwise about (0, 0). Label the image A.

180° about (0, 2). Label the image B.

90° anti-clockwise about (3, 2). Label the image C

90° clockwise about (3,-1). Label the image D.

Q2.

a)

b)

c)

d)

Copy the diagram to the right, and then complete the following.

Rotate A 90° clockwise about (0,0)

Rotate B 180° about (0,0)

Rotate C 90° anti-clockwise about (-1,1)

Rotate D 270° clockwise about (3,0)

Q3. The triangle JKL has coordinates J(-2,-2), K(-2,5) and F(4,5). JKL is rotated 180° about (1,0) to create the image NMO.

Find the coordinates of MNO.

Q4.

a)

b)

c)

d)

e)

Describe the rotation that maps the following:

Q onto R

Q onto S

Q onto T

Q onto U

S onto T

Q5.

a) b)

The triangle ABC has vertices A(1,1), B(3,5) and C(-1,3). The triangle XYZ has vertices X(-2,4), Y(-6,6) and Z(-4,2).

Draw the two triangles on a pair of axes. Describe fully the rotation that maps triangle ABC onto XYZ.

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Translations with Vectors

Q1. Use vector notation to describe the following translations.

i) A to B

ii) A to C

iii) A to D

iv) A to E

v) A to F

vi) A to G

vii) B to E

viii) F to G

ix) C to E

x) G to C

Q2. Translate object A according

to the following vectors.

i) A to B = ( )

ii) A to C = ( )

iii) A to D = ( )

iv) A to E = ( )

v) A to F = ( )

vi) A to G = ( )

Q3.

a)

b)

c)

d)

Draw the following transformations on a copy of the diagram.

Translate kite A by the vector ( ).

Label this new kite B.

Translate kite B by the vector ( ).

Label this new kite C.

Describe, using vector notation, the translation that will map kite A onto C.

Describe, using vector notation, the translation that will map kite C on to A.

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Enlargements on a Grid Q1. Copy each of these diagrams onto squared paper and enlarge them by the scale factor and

centre shown. a) b)

Scale Factor 2 Centre (0, 0) Scale Factor 3 Centre (1, 0)

Q2. Copy the diagrams onto squared paper and by the centre and scale factor given:

Shape A Scale Factor 2 Centre (0,0)

Shape B Scale Factor 3 Centre (-1,2)

Q3. Describe fully the enlargement that maps:

a) A onto A’

b) B onto B’

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SAMPLE

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Negative Scale Factor Enlargements

Q1 Copy each of these diagrams on square paper and enlarge them to the scale factor and centre shown.

a)

i) Scale factor -1 about (1 , 4)

ii) Scale factor -2 about (3 , 6)

b)

i) Scale factor -3 about (-2 , 6)

ii) Scale factor -0.5 about (0 , 7)

Q2 Copy each of these diagrams on square paper and enlarge them to the scale factor and centre shown.

i) Enlarge shape B by a scale factor -2 from (5 , 4). Label the image C.

ii) Describe fully the single transformation that maps A on to B.

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SAMPLE

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Describing Transformations

Q1. Describe fully the single transformations that

will map the shaded triangle onto each of the

triangles A - F.

Q2. Describe fully the single

transformations that will map the

following.

i) A to B

ii) A to C

iii) B to D

iv) A to E

v) C to F

Q3. Describe fully the single transformations

that will map the following.

i) A to B

ii) A to C

iii) A to E

iv) A to F

v) A to G

vi) B to D

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SAMPLE

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Vector Addition Q1. ABCD is a trapezium.

M is the midpoint of AD. N is the midpoint of AC.

Work out the following vectors.

a) b) c) d)

Q2 OAB is a triangle. OBC is a straight line.

M is the midpoint of AB.

Work out in terms of a and b.

Q3.

a)

b)

XYZ is a triangle. OBC is a straight line.

Find in terms of A and B.

G is the point on ZY such that

= 3 : 1

Find in terms of a and b. Give your answer in its simplest form.

Q4.

a)

b)

WXYZ is a parallelogram.

V is on the line SQ such that = 3 : 2

Write down, in terms of a and b, the vector

.

Express in terms of a and b.

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SAMPLE

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Vector Geometry

Q1. Triangle ABE and ACD are similar where AC = 3AB.

and .

Q2. and

,

a) Write down the vectors , , and

in terms of a and b.

a) Write , , and in terms of a and b.

b) What do the vectors show about BE andCD?

b) What do the vectors show about NM and AC?

Q3 PQRA is a quadrilateral.

and

Q4 ABFE is a parallelogram. ABC is a straight line

where AC = 2AB. = a and .

a) Write the vectors , and in terms ofa and b.

a) Write and in terms of a and b.

b) What do the vectors show about thequadrilateral?

b) What do the vectors show about GF and EC?

Q5.

a) b)

OABC is a parallelogram with and . D is a point on OC such that OD:DC = 1:2 and E is a point on AC such that AE:EC = 2:1.

Show that OB is parallel to DE. Show that OB = 3DE.

Q6.

a)

b)

STUV is a quadrilateral. E, F, G and H are mid-points of SV, SU, VT and UT respectively.

, , and

Write vectors , and in terms of a and b.

What type of quadrilateral is EFHG?

15

Reflective  Symmetry  

 

Solutions  

Q1.  

a)    

 

b)     c)    

 

d)    

   

e)    

 

f)    

 

g)     h)    

 

Q2.  

 

No  lines  of  symmetry    

One  line  of  symmetry  

 

Two  lines  of  symmetry  

 

Infinite  lines  of  symmetry.  

 

Q3.  

a)  

   

b)    

 

c)    

 

 

d)    

 

 

 

16

Reflective  Symmetry  

 

Q4.  

a)    

 

b)    

 

 

c)    

 

 

 

 

17

Rotational  Symmetry  

 

Solutions    Q1.  a)  Order  4   b)  Order  2   c)  Order  5   d)  Order  2  e)  Order  2   f)  Order  4   g)  Order  2   h)  Order  2    Q2.  a)  

 

b)    

 

c)    

 

d)    

   Q3.       Square  is  the  odd  one  out  since  the  other  shapes  have  one  order  of  rotational  symmetry  whereas  

the  square  has  four.    Q4.  a)    

 

b)    

 

c)    

   

18

Enlarging Shapes

Solutions

Q1.

a)

b)

c)

d)

e)

f)

Q2.

a)

b)

c)

d)

e)

f)

Q3.

Scale Factor = 2

19

Rotating  Objects  

 

Solutions    Q1.      a)    

 

b)    

 

c)    

 

d)    

 

e)    

 

f)    

 

 Q2.  a)    

 

b)    

 

c)    

 

             

20

Rotating  Objects  

 

Q3.    a)    

 

b)    

 

c)    

       

21

Translating Objects

Solutions

Q1.

a) 2 left, 3 down b) 1 right, 3 down c) 3 down d) 3 left, 4 up

Q2.

a)

b)

c)

d)

e)

f)

g)

h)

Q3. 1 right, 4 up.

Q4. 3 left, 4 down

22

Reflections on a Grid

Solutions

Q1.

Q2.

Q3.

a) W to X Mirror Line � = 0 b) W to Y Mirror Line � = � c) W to Z Mirror Line � = −�

23

Rotations on a Grid

Solutions

Q1.

Q2.

Q3. (4,-2), (4,-5) & (-2,-5)

Q4.

a) Q onto R 180° about (0,0) b) Q onto S 90° CW about (0,-1) c) Q onto T 90° ACW about

(2,0)

d) Q onto U 90° ACW about (5,0) e) S onto T 180° about (0.5,0.5)

Q5. Rotation of 90° anticlockwise about (-2,1)

24

Translations  with  Vectors  

 

Solutions    Q1.  

i)  A  to  B −40   ii)  A  to  C −2

3   iii)  A  to  D 42   iv)  A  to  E 4

−3   v)  A  to  F 0−4  

vi)  A  to  G   −3−4   vii)  B  to  E   8−3   viii)  F  to  G   −30   ix)  C  to  E   6

−6   x)  G  to  C   17  

 Q2.  

   Q3.  

 

c)  A  to  C   52    d)  C  to  A −5

−2  

   

25

Enlargements on a Grid

Solutions

Q1.

a)

b)

Q2.

Q3.

a) A onto A’ Scale Factor 0.5 Centre (2, 0) b) B onto B’ Scale Factor 3 Centre (-1, -2)

26

Negative Scale Factor Enlargements

Solutions

Q1. a)

b)

Q2.

ii) A to B = Enlargement, Scale Factor −1

3, Centre (4 , 6)

27

Describing  Transformations  

 

Solutions    Q1.       A  to  B  Translation   20  

A  to  C  Rotation  180° about (0,0) A to D Translation   −5−3  A  to  E  Rotation  90°  CW  about  (1,1)  A  to  F  Reflection  in  𝑥 = 0    

 Q2.       i)  A  to  B  Rotation  180°  about  (-­‐1  ,  2)  

ii)  A  to  C  Enlargement  SF  2  Centre  (0  ,  0)  iii)  B  to  D  Reflection  in  𝑦 = 1  

iv)  A  to  E  Translation   −1−3  

v)  C  to  F  Rotation  90°  ACW  about  (8  ,  3)    

Q3.        

i)  A  to  B  Translation   −5−1  

ii)  A  to  C  Rotation  90°  CW  about  (3  ,  -­‐1)  iii)  A  to  E  Enlargement  SF  =  -­‐2  Centre  (0  ,  2)  iv)  A  to  F  Enlargement  SF  1 2  Centre  (4  ,  -­‐3)  v)  A  to  G  Rotation  90°  ACW  (2  ,  4)  vi)  B  to  D  Reflection  in  𝑦 = 𝑥      

   

28

Vector Notation

Solutions

Q1.

a) �������� � 3� − 2� b) ��������� ��

� 3� − 2�! c) ������� � 2� − 4� d) � ������� � 2� − �

Q2. ��������� � # − $

Q3. a) �������� = # − $ b) �������� � �

% 5# − $!

Q4. a) �������� � $ − # b) �������� ��

' 2$ + 3#!

29

Vector Geometry

Solutions Q1.

a) 𝐴𝐶 =3a, 𝐴𝐷 = 3𝑏, 𝐵𝐸 = 𝑏 − 𝑎 and 𝐶𝐷 = 3𝑏 − 3𝑎

b) BE and CD are parallel

Q2.

a) 𝑁𝐵 = 3

5𝑏, 𝐵𝐶 = 𝑎 − 𝑏, 𝐵𝑀 =

3

5(𝑎 − 𝑏) and 𝑁𝑀 =

3

5𝑎

b) GF and EC are parallel

Q3.

a) 𝑄𝑅 = 3𝑏 and 𝑅𝑆 = −2𝑎

b) PQRA is a Rhombus

Q4.

a) 𝐺𝐹 =1

2𝑏 − 𝑎 and 𝐸𝐶 = 2𝑎 − 𝑏

b) GF and EC are parallel

Q5. 𝑂𝐵 = 𝑎 + 𝑐, 𝐷𝐸 = 1

3(𝑎 + 𝑐)

Q6.

a) 𝐸𝐹 = 1

2(𝑏 − 𝑎), 𝐸𝐺 =

1

2𝑐 and 𝐹𝐻 =

1

2𝑐

b) EFHG is a parallelogram

30