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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)
3
SAMPLE
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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
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)
6
SAMPLE
SAMPLE
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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.
7
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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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SAMPLE
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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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SAMPLE
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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.
10
SAMPLE
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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’
11
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.
12
SAMPLE
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
13
SAMPLE
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.
14
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
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