© boardworks ltd 2006 1 of 42 ks3 mathematics s5 coordinates and transformations 2

© Boardworks Ltd 2006 1 of 42

KS3 Mathematics

S5 Coordinates and transformations 2


Contents


A

A

A

AS5.1 Translation

S5.2 Enlargement

S5.3 Scale drawing

S5.4 Combining transformations


Translation

When an object is moved in a straight line in a given direction we say that it has been translated.

As an example, we can translate triangle ABC 5 squares to the right and 2 squares up:

C

A

B

object

C

A

B

object

C

A

B

object

C

A

B

object

C

A

B

object

C

A

B

object

C

A

B

object

C

A

B

object C’

A’

B’

image

Every point in the shape moves the same distance in the same direction.

object


Translations

When a shape is translated the image is congruent to the original.

The orientations of the original shape and its image are the same.

An inverse translation maps the image that has been translated back onto the original object.

What is the inverse of a translation 7 units to the left and 3 units down?

The inverse is an equal move in the opposite direction.

That is, 7 units right and 3 units up.


Describing translations

When we describe a translation we always give the movement left or right first followed by the movement up or down.

We can describe translations using vectors.

For example, the vector describes a translation 3 right and 4 down. –4

3

As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left.

A different way of describing a translation is to give the direction as an angle and the distance as a length.


Translations on a coordinate grid

The vertices of a triangle lie on the points A(5, 7), B(3, 2) and C(–2, 6).

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1

Translate the shape 3 squares left and 8 squares down. Label each point in the image.

What do you notice about each point and

its image?

A’(2, –1)

B’(0, –6)

C’(–5, –2)

y

x

C(–2, 6) A(5, 7)

B(3, 2)



The coordinates of vertex A of this shape are (–4, –2).

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1

When the shape is translated the coordinates of vertex A’ are (3, 2).

What translation will map the shape onto its

image?

A’(3, 2)

A(–4, –2)

7 right4 up

y

x



The coordinates of vertex A of this shape are (3, –4).

When the shape is translated the coordinates of vertex A’ are(–3, 3).

What translation will map the shape onto its

image?

6 left7 up

1 2 3 4 5 6–2–3–4–5–6–7

1

2

5

6

–2

–4

–6

–3

–5

–7

–1

y

x7–1

3

4

7

0

A(3, –4)

A’(–3, 3)


Translation golf


Contents


A

A

A

A

S5.2 Enlargement

S5.1 Translation

S5.3 Scale drawing



Find the missing lengths

The second photograph is an enlargement of the first.What is the length of the missing side?

4 cm

3 cm

10 cm

3 cm ?7.5 cm



The second photograph is an enlargement of the first.What is the length of the missing side?

?

5 cm12.5 cm

10 cm

4 cm


6.7 cm

5.8 cm

?

?


The second picture is an enlargement of the first picture.What are the missing lengths?

5.6 cm

11.2 cm

2.9 cm

13.4 cm6.7 cm

5.8 cm



The second shape is an enlargement of the first shape.What are the missing lengths?

4 cm

6 cm

6 cm

5 cm

3 cm9 cm

7.5 cm

4.5 cm

?

?

?

4 cm

4.5 cm

5 cm



The second cuboid is an enlargement of the first.What are the missing lengths?

1.8 cm

5.4 cm

1.2 cm

3.5 cm10.5 cm

3.6 cm

?

?

3.5 cm

3.6 cm


Enlargement

AA’

Shape A’ is an enlargement of shape A.

The length of each side in shape A’ is 2 × the length of each side in shape A.

We say that shape A has been enlarged by scale factor 2.


Enlargement

When a shape is enlarged the ratios of any of the lengths in the image to the corresponding lengths in the original shape (the object) are equal to the scale factor.

A

B

C

A’

B’

C’

= B’C’BC

= A’C’AC

= the scale factorA’B’AB

4 cm6 cm

8 cm

9 cm6 cm

12 cm

64

= 128

= 96

= 1.5


Congruence and similarity

Is the image of an object that has been enlarged congruent to the object?

Remember, if two shapes are congruent they are the same shape and size. Corresponding lengths and angles are equal.

In an enlarged shape the corresponding angles are the same but the lengths are different.

The image of an object that has been enlarged is not congruent to the object, but it is similar.

In maths, two shapes are called similar if their corresponding angles are equal. Corresponding sides are different lengths, but the ratio in lengths is the same for all the sides.


Find the scale factor

What is the scale factor for the following enlargements?

B

B’

Scale factor 3




Scale factor 2

C

C’




Scale factor 3.5

D

D’




Scale factor 0.5

E

E’


Using a centre of enlargement

To define an enlargement we must be given a scale factor and a centre of enlargement.

Here is how to enlarge triangle ABC by scale factor 2 from the centre of enlargement O:

O

A

CB

OA’OA

= OB’OB

= OC’OC

= 2

A’

C’B’


Using a centre of enlargement

Enlarge parallelogram ABCD by a scale factor of 3 from the centre of enlargement O.

O

DA

BC

OA’OA

= OB’OB

= OC’OC

= 3= OD’OE

A’ D’

B’ C’


Exploring enlargement


Enlargement on a coordinate grid

The vertices of a triangle lie on the points A(2, 4), B(3, 1) and C(4, 3).

The triangle is enlarged by a scale factor of 2 with a centre of enlargement at the origin (0, 0).

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

A(2, 4)

B(3, 1)

C’(8, 6)

A’(4, 8)

B’(6, 2)


its image?

y

x

C(4, 3)


Enlargement on a coordinate grid

The vertices of a triangle lie on the points A(2, 3), B(2, 1) and C(3, 3).

The triangle is enlarged by a scale factor of 3 with a centre of enlargement at the origin (0, 0).


its image?0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10y

x

A(6, 9) C’(9, 9)

B’(6, 3)

A(2, 3)

B(2, 1)

C(3, 3)


Contents


A

A

A

A

S5.3 Scale drawing

S5.1 Translation

S5.2 Enlargement



Scale drawings

We use scale drawings to represent real objects drawn in proportion to their actual sizes.

If we are given a scale for a picture then we can work out the size of an object in real life.

Below is a scale picture of a 10p coin.

0.5 cm in this picture represents 1 mm in real life.

The coin in the picture has a diameter of 12.2 cm. What is the

actual diameter of the coin?

The actual diameter is 24.4 mm.


Scale drawings

Here is a scale drawing of a car.

Every 1 cm in this drawing represents 50 cm in real life.

If the length of the car in the drawing is 4.5 cm, what length is the car in real life?

Length of the car in real life = 4.5 × 50

= 225 cm

= 2.25 m


Drawing a plan

On maps and plans, the scale is usually given as a ratio.

For example, Frank decides to draw a plan of his bedroom using a scale of 1 : 20.

That means that every 1 cm in the plan represents 20 cm or 0.2 m in real life.

He measures his room to find that it has a length of 360 cm and a width of 250 cm.

What will the length and the width of the room be in the scale drawing?

Length = 360 ÷ 20 = 18 cm

Length = 250 ÷ 20 = 12.5 cm


Drawing a plan

Frank uses a table to convert between the sizes of the things in his room and their sizes in his plan:

Object Actual size Size in the plan

Width of door 80 cm

Bed 90 cm by 190 cm

Chest of drawers 68 cm by 52 cm

Wardrobe 6 cm by 2.5 cm

Desk 1.8 cm by 3.2 cm

Bookshelf 1.75 cm by 3.9 cm

4 cm

4.5 cm by 9.5 cm

3.4 cm by 2.6 cm

120 cm by 50 cm

36 cm by 64 cm

35 cm by 78 cm


Maps

A map uses a scale of 1 : 40 000.

How many km are represented by 1 cm on the map?

1 cm on the map is 40 000 cm in real life.

40 000 cm = 400 m = 0.4 km

Two towns are 3.5 cm apart on the map.How far apart are they in real life?

3.5 cm × 0.4 = 1.4 km

1 cm on the map is 0.4 km in real life.


Maps

A map uses a scale of 1 : 50 000.

How many km are represented by 1 cm on the map?

1 cm on the map is 50 000 cm in real life.

50 000 cm = 500 m = 0.5 km

Two towns are 2.3 km apart in real life.How far apart are they on the map?

2.3 km ÷ 0.5 = 4.6 cm

1 cm on the map is 0.5 km in real life.


Contents


A

A

A

A


S5.1 Translation

S5.2 Enlargement

S5.3 Scale drawing


Combining reflections

An object may be reflected many times.

In a kaleidoscope mirrors are placed at 60° angles.

Shapes in one section are reflected in the mirrors to make a pattern.

How many lines of symmetry does the

resulting pattern have?

Does the pattern have rotational symmetry?


Parallel mirror lines

What happens when an object is reflected in parallel mirror lines placed at equal distances?


Parallel mirror lines

Reflecting an object in two parallel mirror lines is equivalent to a single translation.Reflecting an object in two parallel mirror lines is equivalent to a single translation.

M1 M2

A A’ A’’

Suppose we have two parallel mirror lines M1 and M2.

We can reflect shape A in mirror line M1 to produce the image A’.

We can then reflect shape A’ in mirror line M2 to produce the image A’’.

How can we map A onto A’’ in a single transformation?


Perpendicular mirror lines

M2

M1

A A’

A’’

We can reflect shape A in mirror line M1 to produce the image A’.

We can then reflect shape A’ in mirror line M2 to produce the image A’’.

How can we map A onto A’’ in a single transformation?

Reflection in two perpendicular lines is equivalent to a single rotation of 180°.Reflection in two perpendicular lines is equivalent to a single rotation of 180°.

Suppose we have two perpendicular mirror lines M1 and M2.


Combining rotations

Suppose shape A is rotated through 100° clockwise about point O to produce the image A’.

O

A

A’

100°Suppose we then rotate shape A’ through 170° clockwise about the point O to produce the image A’’.

How can we map A onto A’’ in a single transformation?170°

A’’

To map A onto A’’ we can either rotate it 270° clockwise or 90° anti-clockwise.

Two rotations about the same centre are equivalent to a single rotation about the same centre.

Two rotations about the same centre are equivalent to a single rotation about the same centre.


Combining translations

Suppose shape A is translated 4 units left and 3 units up.

Two or more translations are equivalent to a single translation.

Two or more translations are equivalent to a single translation.

A

A’’

Suppose we then translate A’ 1 unit to the left and 5 units down to give A’’.

A’

How can we map A to A’’ in a single transformation?

We can map A onto A’’ by translating it 5 units left and 2 units down.


Transformation shape sorter

© boardworks ltd 2006 1 of 42 ks3 mathematics s5 coordinates and transformations 2

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