29 plane isometric transformationswestsidemathematics.wikispaces.com/file/view/ch29.pd… · ·...

510

Using mathematics: real-life applications

You can see examples of reflections, rotations and translations all around you. Patterns in wallpaper and fabric are often translations, images reflected in water are reflections and the blades of a wind turbine are a good example of rotation.

Before you start …

Ch 9 You need to know what angles of 90°, 180° and 270° look like and also the directions clockwise and anti-clockwise.

1 How many degrees is each angle? State whether each arrow is showing clockwise or anti-clockwise movement.

a b c

Ch 18, 19

You need to know how to plot straight-line graphs in the form x 5 a, y 5 a and y 5 x.

2 Draw the graph for each equation.

a y 5 2 b y 5 x

c x 5 21 d y 5 2x

Ch 27 You need to know what a vector is and how they describe movement.

3 a What is the difference between the coordinate (3, 2) and the

vector ( 3 2

) ?

b What is the relationship between the vectors ( 21 3 ) and ( 3

1 ) ?

Tracing paper is very useful for work with transformations. Don’t be afraid to ask for it in an exam.

Tip

“I use transformations all the time when I program computer graphics. Transformations allow me to position objects, shape them and change the view I have of them. I can even change the type of perspective that is used to show something.” (Computer programmer)

29 Plane isometric transformations

For more resources relating to this chapter, visit GCSE Mathematics Online.

In this chapter you will learn how to …

• carry out rotations, reflections and translations.

• identify and describe rotations, reflections and translations.

• describe translations using column vectors.

• perform multiple transformations on a shape and describe the results.

© Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms


511Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

Assess your starting point using the Launchpad

Go toSection 1: Reflections

Go toSection 2: Translations

✓

1 Reflect the given shape in the mirror line.

STEP 1

2 What is the equation of the mirror line?

y

x0

�4

�5

�3

�2

�1

1

2

3

�2 �1 1 2 3 4 5

STEP 2

3 Translate the shape

through the vector ( 3 2

) .

4 Describe the translation of this object to its image.

y

x0

�4

�3

�2

�1

1

2

3

�2�3�4�5 �1 1

image

object

2

✓

Go toStep 3: The Launchpad continues on the next page …

512

GCSE Mathematics for OCR (Higher)

Section 1: ReflectionsA transformation is a change in the position of a point, line or shape. When you transform a shape you change its position or its size.

The original point, line or shape is called the object. For example, the triangle ABC.

The transformation is called the image. The symbol 9 is used to label the image. For example, the image of triangle ABC is A9B9C9.

Reflection, rotation and translation change the position of an object, but not its size. Under these three transformations an object and its image will be congruent.

Enlargement is also a transformation. Enlargement changes the position of an object and also its size. Under enlargement an object and its image are similar. You will learn about similarity in more detail in Chapter 30.

Mirrors, windows and water surfaces all reflect objects. You can see the reflection of clouds and trees clearly in the photograph. If you draw a line horizontally across the centre of the image and fold it, the top half will fit exactly onto the bottom half. The fold line is called a mirror line.

Mathematically, when a shape is reflected it is flipped over a mirror line to give its image. The object and the image are the same distance from the mirror line.

object: the original shape (before it has been transformed).

image: the new shape (once the object has been transformed).congruent: shapes that are identical in shape and size.

similar: shapes that have the same shape and proportions but are a different size.

mirror line: a line equidistant from all corresponding points on a shape and its reflection.

Key vocabulary

You learnt about reflection in Chapter 5, and will learn more about congruency in Chapter 29.

Tip

Go toSection 3: Rotations

Go toSection 4: Combined transformations

Launchpad continued …

✓

Go toChapter review

STEP 3

5 Rotate the shape 90° anticlockwise around (1, 1).

STEP 4

6 Reflect the shape O in the y-axis and label the image O9.

Rotate O9 90° anti-clockwise and label the image O0.

What single transformation maps O to O0?

y

x0�1

1

2

3

�2 �1 1

y

x0�11

1

22

33

��2 ��1 11

1

✓




EXERCISE 29A

1 Reflect the triangle in the line x 5 1, and then reflect the triangle and the resultant image in the line y 5 21.

2 Reflect this shape in the line y 5 x, y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

and then reflect the shape and the resultant image in the line y 5 1 2 x.

3 Carry out the ten reflections on the grid below to reveal the word.Shape A in the line y 5 x Shape B in the x-axis Shape C in the line y 5 22 Shape D in the line y 5 4 Shape E in the line x 5 7 Shape F in the line y 5 2xShape G in the line x 5 24Shape H in the line x 5 1.5Shape I in the line y 5 6Shape J in the line y 5 x

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

y

x0�4�5�6�7�8�9�10

�4

�5

�6

�7

�8

�9

�10

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

�3 �2 �1 1 2

J

C

B

F

G D

I

A

EH

3 4 5 6 7 8 9 10

514


4 Draw a pair of axes going from 210 to 10 in both directions. Draw two line segments the first with ends at (0, 0) and (1, 1) the second with ends at (1, 1) and (2, 1). Show how these lines can be made into:

a a hexagon with two successive reflections.

b an octagon with three successive reflections.

5 Make up your own reflection puzzle for another student to solve.

Describing reflections

You need to be able to draw a mirror line on a diagram. You must also be able to give the equation of the mirror line when the reflection is shown on a coordinate grid. This is known as describing the reflection.

The mirror line is the perpendicular bisector of two corresponding points in a reflection. So, if you can’t ‘spot’ a mirror line, you can join two corresponding points and construct the perpendicular bisector to find it.

EXERCISE 29B

1 Find the equation of the mirror line in each reflection.

To check a reflection, trace the object, the image and the mirror line. Fold the tracing paper along the mirror line; the shapes should match up exactly.

Tip

a y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

b y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

c y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

d y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5




2 a Fully describe each of the following reflections.

i Shape A to shape E.

ii Shape C to shape G.

iii Shape G to shape E.

iv Shape B to shape F.

b Challenge another student to describe a reflection of two triangles you choose.

3 Trace each pair of shapes and construct the mirror line for the reflection.

a b c

4 a In each diagram below, describe the multiple reflections that take the original shaded shape to its image.

i y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

ii

b Is there more than one answer?

c Does the order of the reflections matter?

d Is there any easy way to tell the minimum number of reflections needed?

y

x0

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

11

12

�2 �1 1 2 3

BA

H

G

FE

D

C

4 5 6 7 8 9 10 11 12 13 14

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

516


Section 2: TranslationsA translation is a slide along a straight line. (Think about pushing a box across a floor.) The translation can be from left to right (horizontal), up or down (vertical) or both (horizontal and vertical).

The image is in the same orientation as the object and every point on the shape moves exactly the same distance in exactly the same direction. Translated shapes are congruent to each other.

You can describe translations on a coordinate grid using column vectors. Remember, a column vector shows horizontal displacement over vertical displacement.

WORKED EXAMPLE 1

Describe the translation ABC to A9B9C9 by means of a column vector. y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3

C9

B9A9A

C

B

4 5

y

x0�4�5

1

2

3

4

5

�3 �2 �1 1 2 3

C9

B9A9A

C

B

4 5

1

2

The translation is ( 2 21

) .

Write this as a column vector.

Work out how the point has been translated horizontally and vertically. To get from C to C9 you move:2 units to the right 5 121 unit down 5 21

Take any point on the object and find the corresponding point on the image. Look at point C and point C9.

orientation: the position of a shape relative to the grid.

Key vocabulary

You learnt about vectors in Chapter 28.

Tip

Drawing on the grid to show the movements can help you to avoid unnecessary mistakes.

Tip




This shape is translated through a vector of ( 22 4 ) . Draw its image.

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

Which one of these answers is correct? What has gone wrong in each of the others?

option A option B option Cy

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

WORK IT OUT 29.1

EXERCISE 29C

1 Translate each shape as directed.

a Translate 6 right and 2 up. b Translate 3 right and 1 down. c Translate 2 down and 4 right.

518


2 Translate each shape using the given vector.

3 Translate each shape by the given vector, then give the name of the shape you have put together.

a Translate shape A by ( 21 23 ) .

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

A

B

C

b Translate shape B by ( 1 5

) .

c Translate shape C by ( 2 21

) .

4 Translate each piece of this jigsaw using the vectors on the right.

A

D B

H

G

F

E

I

C

A ( 12 28

)

B ( 12 25

)

C ( 10 29

)

D ( 14 24

)

E ( 7 29

)

F ( 3 27

)

G ( 13 25

)

H ( 5 22

)

I ( 21 22 )

a ( 3 21

) b ( 21 2 ) c ( 0

4 )




Describing translations

You should be able to use vectors to describe a translation. Remember to count between corresponding points on the two shapes.

Which transformations below are reflections and which are translations? How did you make your decision?


x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3

image

4 5

object

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3

image

4 5

object

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3

image

4 5

object

WORK IT OUT 29.2

EXERCISE 29D

1 Here are some completed translations. The objects (A to E) are shown in colour and the images are in grey. Write column vectors to describe the translation from each object to its image.

y

x0�4�5�6�7�8�9�10

�4

�5

�6

�7

�8

�9

�10

�3

�2

�1

1

A

B

C

D E2

3

4

5

6

7

8

9

10

�3 �2 �1 1 2 3 4 5 6 7 8 9 10

A9

E9

B9

C9

D9

Make sure you count from the object to the image and write this as a column vector (not a coordinate)!

Tip

520


2 Work on a square grid. Draw the four objects (A to D) used to make this image in any position on the grid.

Make up translation instructions for moving the four objects to form the image.

Exchange with a partner and perform the translations to make sure their instructions are correct.

Section 3: RotationsA rotation is a turn. An object can turn clockwise or anti-clockwise around a fixed point called the centre of rotation. The centre of rotation may be inside, on the edge of, or outside the object.

A rotation changes the orientation of a shape, but the object and its image remain congruent.

When you rotate a shape, the distance from the centre of rotation to any point on the object remains the same. Each point travels in a circle around the centre. Think about the tip of a blade on a wind turbine or a child sitting on a roundabout. When they rotate, the paths they trace out are circles.

To carry out a rotation you need to know the centre of rotation as well as the angle and direction of rotation. At this level, all rotations will be in multiples of 90°.

This shape is rotated anti-clockwise, with centre of y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

rotation (0, 1) through an angle of 90°. What is its image?

Which one of these answers is correct? What has gone wrong in each of the others?


x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

WORK IT OUT 29.3

D

C

A

B

centre of rotation




EXERCISE 29E

1 Rotate the triangle 180° about the origin.

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

2 Rotate the shape 90° clockwise around the point (1, 1).

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

3 Rotate the shape 90° anti-clockwise around the point (22, 1).

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

522


4 Rotate the shape 90° anti-clockwise y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

around the point (2, 1).

5 Rotate each shape as directed.

Shape A: 90° anti-clockwise around the point (21, 21).

Shape B: 180° around the point (2, 3).

Shape C: 90° clockwise around the point (1, 0). Label this D.

Shape D: 180° around the point (23.5, 2).

Shape E: 180° around the point (3, 1).

6 This image was designed by drawing y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

a triangle and rotating this around the origin in multiples of 90°.

What do you notice about the coordinates of the vertices of the triangle? Would this work if you rotated an image around a different point? Why?

Describing rotations

To describe a rotation you give a centre, angle and direction. Very often you can find the centre using tracing paper and trial and error. Trace the object and rotate the tracing paper using different centres of rotation. Spotting the centres improves with practice.

You can also use construction to find the centre of rotation. Corresponding points on an object and its image under rotation lie on the circumference of a circle. If you join the corresponding points to make a chord, the perpendicular bisector of the chord will go through the centre of the circle.

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4

E

C

B

A

5

Always give the direction and angle of the rotation from the object to the image.

Tip




So by drawing the perpendicular bisectors of two chords you can determine the centre.

centre of rotation

The broken lines show you the circle and chord to help demonstrate the principle; you do not need to draw these!

Which of these transformations are reflections, which are rotations and which are translations? How did you make your decision? Were there any transformations you couldn’t identify?


x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

option D option E option Fy

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2

image

object

3 4 5

WORK IT OUT 29.4

524


EXERCISE 29F

1 Describe each of the following rotations.

2 This section of wallpaper has been designed using rotations. A coordinate grid has been overlaid.

Identify as many different rotations as you can.

3 Use construction to locate the centres of rotation for each pair of shapes.

a y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3

image

object

4 5

b y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

image

object

c y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

image

object

d y

x0�4�5

�4

�5

�3

�2

�1

1

2

3

4

5

�3 �2 �1 1 2 3 4 5

image

object

y

x0

�10

�5

10

5

�10 105�5

�10

�5

10

5




Section 4: Combined transformations

You have seen that an object can undergo a single transformation to map it onto an image. An object can undergo two (or more) transformations in a row. The transformations do not have to be of the same type. For example, an object could be reflected in the line y 5 0 and then rotated clockwise through 90° about a vertex. Sometimes a combined transformation can be described by a single, equivalent transformation.

EXERCISE 29G

1 On your copy of this diagram, reflect the object A in the line x 5 2 and label it A9.

Rotate shape A9 90° anti-clockwise around the point (2, 22) label it A0.

Describe the single transformation that maps shape A to shape A0.y

x0�4�5�6

�4

�5

�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

A

2 On your copy of this diagram, translate F through a vector of ( 23 2 ) and

label it F9.

Rotate F9 180° around the point (23, 0) and label if F0.

Describe the single transformation that maps shape F onto to shape F0.y

x0�4�5�6

�4

�5

�6

�3

�2

�1

1

2

3

F

4

5

6

�3 �2 �1 1 2 3 4 5 6

526


3 On your copy of this diagram, reflect K y

x0�4�5�6

�4

�5

�6

�3

�2

�1

1

2

3

K

4

5

6

�3 �2 �1 1 2 3 4 5 6

in the y-axis label it K9.

Rotate K9 90° clockwise around the point (2, 21) and label it K0.

Reflect K0 in the line x 5 0, label it K-.

Describe the single transformation that maps shape K onto shape K-.

4 a Describe fully the combined y

x0�4�5�6

�4

�5

�6

�3

�2

�1

1

2

3

4

5A9

A

6

�3 �2 �1 1 2 3 4 5 6

transformation from shape A to A9.

b Does the order of the transformations matter?

c Find another solution.

5 A shape is transformed by carrying out a reflection in the x-axis followed by a reflection in the y-axis. What single transformation has the same effect?

6 What two transformations would y

x0�4�5�6

�4

�5

�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

have the same effect as rotating the following shape 90 degrees anti-clockwise around the point (22, 0)?




Checklist of learning and understanding

Transformations

A transformation is a change in the position of a point, line or shape. When you transform a shape you change its position or its size.

The original point, line or shape is called the object: for example, triangle ABC.

The transformation is called the image. The symbol 9 is used to label the image. For example, the image of triangle ABC is A9B9C9.

Different types of transformation include reflections, translations, enlargements and rotations.

Reflections

Reflections change the orientation of a shape but the image remains congruent.

To describe a reflection the equation of the mirror line needs to be given.

The mirror line is the perpendicular bisector of the line linking any two corresponding points on the image and object.

Translations

Translations leave the orientation of the shape unchanged but move it horizontally and/or vertically.

Translations are described using vectors.

Rotations

A rotation is a turn around a centre. Rotations are described by giving the coordinates of the centre of rotation, angle and direction of the rotation.

Combined transformations

Transformations can be combined by completing one after another; the transformations do not have to be of the same type. Sometimes the result of combined transformations can be achieved with a single, equivalent transformation.

Chapter review

1 Which of the following statements are true? Explain your reasoning

a The images constructed by reflecting, rotating, or translating are congruent to the objects you started with.

b The images constructed by reflecting, rotating, or translating are similar to the objects you started with.

c The images constructed by reflecting, rotating, or translating are in the same orientation to the objects you started with.

d The images constructed by reflecting, rotating, or translating have the same angles as the objects you started with.

For additional questions on the topics in this chapter, visit GCSE Mathematics Online.

528


2 Describe fully the transformation y

x0�4�5�6

�4�5�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

A

BC

from:

a shape A to shape B.

b shape B to shape C.

c shape C to shape A.

3 This triangle is used to create y

x0�4�5�6

�4�5�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

a tessellating pattern. This is produced using multiple translations and one rotation. Explain how this could be done.

4 y

x0�4�5�6

�4�5�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

A

Rotate shape A 90° clockwise around the point (1, 2). Label it B.

Reflect shape B in the line y 5 x. Label it C.

Translate shape C through the vector ( 21 1 ) . Label it D.

Describe the single transformation that maps shape A onto shape D.

Tessellation is a pattern of shapes that fit together perfectly, with no gaps or overlaps.

Tip




5 Look at the dancing figure below. Describe how the figure can be drawn using only transformations of shapes A, B, C and D.

y

x0�4�5�6

�4�5�6

�3

�2

�1

1

2

3

4

5

6

�3 �2 �1 1 2 3 4 5 6

A B

D

C

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