Mathematics

Key unit competence: By the end of this unit, learners should be able to solve
   shape problems involving inverse and composite transformations.

Unit outline

• Composite translations in two dimensions

• Composite reflections in two dimensions

• Composite rotations in two dimensions

• Mixed transformations in two dimensions

• Inverse transformations two dimensions.

Introduction

Unit Focus Activity

ABCD is a trapezium whose vertices are
  A(1 , 2), B(7 , 2), C(5 , 4) and D(3 , 4).

(a) On the same grid:

(i) Draw ABCD and its image A′B′C′D′ under a rotation of
     90° clockwise about the origin.

(ii) Draw the image A′′B′′C′′D′′ of A′B′C′D′ under a reflection
     in the line y = x. State the co-ordinates of A′′B′′C′′D′′.

(b) A′′′B′′′C′′′D′′′ is the image of A′′B′′C′′D′′ under the reflection
     in the line y = 0. Draw the image

A′′′B′′′C′′′D′′′ and state its co-ordinates.

(c) Describe a transformation that maps A′′′B′′′C′′′D′′′ onto ABCD.

We have also learnt how to perform single transformations including reflection,
rotation and translation. In this unit, we will discover how to perform composite
transformation i.e one transformation followed by another and so on.

12.1 Composite translations in two dimensions A B C

Activity 12.1

Given the vertices of a triangle ABC
as A (–4, 3), B (–1, 1) and C (–2, 5).

(a) Plot the points ABC on a
Cartesian plane

(b) Join the points A, B and C to form triangle ABC

(c) Move point A horizontally to the right by 6 units and then
    vertically upwards by 2 units. Locate the final position of A as A′.

(d) Repeat procedure (c) for points B and C.

(e) Join the points A′, B′ and C′

(f) What do you observe about the size and shape of ABC and A′B′C′?

(g) Move point A′ 4 units downwards along the grid. Then locate the
     final position A′′. Repeat this procedure for both points B′and C′.

Composite transformation takes place when two or more transformations combine
to form a new transformation. Here, one transformation produces an image upon which
the other transformation is then performed. In a translation, all the points in the object
move through the same distance and in the same direction. In order to obtain the
image of a point under a translation, we add the translation vector to the position
vector of the point.

12.2 Composite reflections in two  dimensions
    Activity 12.2

1. Given the vertices of triangle ABC as A(-2, 1), B(-5,1) and C(-4,-3).

(a) Draw ABC, line x=0 and line
     y=3 on the Cartesian plane.

(b) Reflect triangle ABC into the
      line x=0 to form the image A′B′C′. Write down the coordinates
      of the image.

(c) Reflect A′B′C′ into the line
     y=3 to k form the image A′′B′′C′′. Write down the
     co-ordinates of the image.

A composite reflection is two or more reflection performed one after the other.
Some simple reflections can be performed easily in the co-ordinate plane using the
following general rules.

(a) Reflection in x-axis Consider point P(2, 3) in the graph and
    assume x-axis is the mirror. The image of P under the reflection is P′(2, –3). The
    x-axis value does not change but the y-axis value changes the direction.

The rule for the reflection in the x-axis is P(x, y ) → (x, – y).
(b) Reflection in the y-axis Consider point in the graph 12.4 and
    assume axis is the mirror. The image of P(–2, 3). The y-axis value does not change
    but the y-axis value changes the direction.

The rule for the reflection in the y-axis is P′(x, y) → P′(–x, y).
(c) Reflection in line y = x Consider point P(–3, 2) in the graph12.5
below and y = x assume is the mirror. The image of P under the reflection is P′(2, –3)
the -axis value becomes -axis value whileaxis value becomes-axis value.

The rule for the reflection in the line is y = x is P(x, y) → P′(y, x).

(d) Reflection in the line y = –x Consider point P in the graph 12.6 and
assume line y = –x is the mirror. The image of P (3, 2) under the reflection is
the x-axis value becomes –y-axis value while y-axis value becomes –x-axis value
ie P′ (–2, –3).

The rule for a reflection in the line y = –x is P(x, y) → P′(–y, –x).

(e) Equation of the mirror line

Equation of the mirror line can be found by using the coordinates of the midpoints
of the image and the object lines.

Example 12.3

The points A (1, 3), B (1, 6) and C (3, 6) are the vertices of triangle ABC.

(a) On a Cartesian plane, draw triangle ABC.

(b) Triangle ABC is reflected in the line y = x. Draw the image A’B’C’
      on the same Cartesian plane and state its co-ordinates.

(c) Triangles A′B′C′ is then reflected in the line y = 0 (x-axis). Draw
     the image A′′B′′C′′ of A′B′C′ on the same Cartesian plane and state
     its co-ordinates.

Co-ordinates of the images. A’ (3,1),B’ (6,1),C’ (6,3)
   A′′(3,–1), B′′ (6,–1), C′′ (6, –3)

Example 12.5

The vertices of a quadrilateral are A(2, 0.5), B(2, 2), C(4, 3.5) and D(3.5,
1). Find the image of the quadrilateral
   under reflection in line y=0 followed by reflection in line y = –x.

Solution

‘Reflection in line y=0 followed by reflection in line y=–x means that we
first obtain the image under reflection in line y=0 and then reflect this image
 in line y = –x.

This is shown in graph 12.9. In the graph, A′B′C′D′ is the reflection of
ABCD in line y = 0. A′′B′′C′′D′′ is the reflection of A′B′C′D′ in line y = –x.
Thus the required image vertices are: A′(2, –0.5), B′(2, –2), C′(4, –3.5) and
D′(3.5, 1) A′′(0.5, –2), B′′(2, –2), C′′(3.5, –4) and D′′(1, 3.5)

Example 12.6

In graph 12.10 triangle ABC has vertices A(–3,3), B(–3,1) and C(–2,2).
A′B′C′ and A′′B′′C′′ are its images after given reflections.

(a) Write down all image co-ordinates.

(b) Write down equations of the reflection lines.

Solution

(a) Image co-ordinates.
     A′(3,3),B′ (3,1),C′(2,2)
     A′′(3, –3) B′′(3, –1), C′′(2, –2)

(b) Equations of refrection lines.
     From triangle ABC to A′B′C′,
     equation of the line x=0.
    From triangle A′B′C′ to A′′B′′C′′,
    equation of the line is y=0.

Exercise 12.2

1. (a) Draw triangle ABC with vertices as
    A (6, 8), B (2, 8), C (2, 6).
   Draw the lines y = 2 and y = x.

(b) Draw the image of triangle ABC after reflection in;

(i) The y-axis. Label it triangle A′B′C′.

(ii) The x-axis. Label it triangle A′′B′′C′′.

(iii) The line y=x. Label it triangle A′′′B′′′C′′′.

(c) Write down the co-ordinates of the image of point A in each case.

2. (a) Plot triangle 1 defined by (3,1), (7,1) (7,3).

   (b) Reflect triangle 1 in the line y = x onto triangle 2.

(c) Reflect triangle 2 in the line y = 0 (x-axis) onto triangle 3.

(d) Reflect triangle 3 in the line
     y = -x onto triangle 4.

(e) Reflect triangle 4 in the line x = 2 onto triangle 5.

(f) Write down the co-ordinates of triangle 5.

3. (a) Construct triangle 1 at (2, 6), (2, 8), (6, 6).

(b) Reflect triangle 1 in the line
     x + y = 6 onto triangle 2

(c) Reflect triangle 2 in the line x = 3 onto triangle 3.

(d) Reflect triangle 3 in the line
     x + y = 6 onto triangle 4

(e) Reflect triangle 4 in the line
     y = x - 8 onto triangle 5.

(f) Write down the co-ordinates of triangle 5.

4. (a) Draw and label the following triangles.

Triangle 1: (3, 3), (3, 6), (1, 6)

Triangle 2: (3,–1), (3, –4), (1, –4)

Triangle 3: (3, 3), (6, 3), (6, 1)

Triangle 4: (–6, –1), (–6, –3),
(–3, –3)
Triangle 5: (–6, 5), (–6, 7), (–3, 7)

(b) Find the equation of the mirror line for the reflection;

(i) Triangle 1 onto triangle 2

(ii) Triangle 1 onto triangle 3

(iii) Triangle 1 onto triangle 4

(iv) Triangle 4 onto triangle 5.

5. Rectangle ABCD has vertices

A(–1,0), B(–1,2), C(1,0) and D(1,2).
(a) Construct rectangle ABCD on a Cartesian plane.

(b) Reflect rectangle ABCD in thex-axis.

(i) Write down the imagecoordinates.

(ii) Construct the image of ABCD on the same
     Cartesian plane.

12.3 Composite rotations in two dimensions
   Activity 12.3

1. You are given a line segment with end points A (2, 2) and B (2, 5).

(a) Draw line AB on a Cartesian plane.

(b) Fix point A and rotate pointB through 60º in clockwise direction.

(c) Write down the co-ordinate B′ the image of B.

(d) Leaving A as a fixed point, rotate point B′ 90º in
    anticlockwise direction.

(e) Write down the co-ordinate B′′ the image of B′.

2. Fig.12.1 shows unmarked clock.

(a) Mark the clock with appropriate time values if
the minute and hour hands are moving in clockwise direction.

(b) Mark the clock with appropriate time values if
     the minute and hour hands are moving in anti-clockwise
     direction.

The three important factors of rotation are direction of rotation, Centre of
rotation and the angle of rotation. The angle of rotation is measured in
degrees. A rotation of 3600 is called a

revolution. A rotation of 270° is the equator turn. A rotation of 180° is called
a half turn and a rotation of 90° is called a quarter turn. Figure 12.11 shows the
various turns. The dotted lines show the initial positions.

The direction in which the hands of a clock turn is called clockwise and the
rotation in the opposite direction is called anticlockwise.
The angle of rotation in an anticlockwise direction is positive and that in a
clockwise direction is negative. Thus, a rotation of 90º anticlockwise is
written as +90º and 90º clockwise as -90º.

When an object is given a rotational transformation, the image is always
the same size as the object. Such a transformation is called an isometry.

If M(h,k) is any point, its rotation follows the following rules.

1. M(h,k) has image M′(–k,h) after rotation at 90° in anticlockwise
   direction.

2. M(h,k) has the image M′(k,–h) after rotation at 90° in clockwise direction.

3. M(h,k) has the image M′(–h, –k) after rotation at 180° in either clockwise or
    anticlockwise direction.

Co-ordinates of the Images.
A′(–2, 3), B′ (–4, 3), C′(–2, 5)
A′′(–3,–2), B′′ (–3,–4), C′′ (–5,–2)

(d) A′(1, 3), B′(4, 1), C′(2, 5)
     A′′(3, –1), B′′(1, –4) C′′(5, –2)
    A′′′(–1, –3), B′′′(–4, –1),
    A′′′(–1, –3) C′′′(–2, –5),

Exercise 12.3

1. (a) Draw triangle one at A(1,2),  B(1,6), C(3,5).

   (b) Rotate triangle one 90o clockwise centre (1,2) onto triangle two.

  (c) Rotate triangle two 180o centre (2,-1) onto triangle three.

  (d) Rotate triangle three 90o clockwise, centre (2,3) onto triangle four

(e) Write down the co-ordinates of triangle four.

2. (a) Draw and label the following triangles on a Cartesian plane.

     Triangle 1: (3, 1), (6, 1), (6, 3)

    Triangle 2: (-1, 3), (-1, 6), (-3, 6)

    Triangle 3: (1, 1), (-2, 1), (-2,-1)

    Triangle 4: (-3, 1), (–3,4), (–5,4)

  (b) Describe fully the following  rotations.

   (i) 1 onto 2            (ii) 1 onto 3

   (iii) 1 onto 4          (iv) 3 onto 2

3. (a) Draw triangle 1 at (4,7), (8,5), (8,7).

   (b) Rotate triangle 1, 90° clockwise centre (4, 3) onto triangle 2.

   (c) Rotate triangle 2, 180° centre (5, –1) onto triangle 3.

  (d) Rotate triangle 3, 90° anticlockwise, centre (0, –8) onto triangle 4.

(e) Describe fully the following  rotations;

   (i) Triangle 4 onto triangle 1

   (ii) Triangle 4 onto triangle 2.

4. (a) Draw triangle LMN at L(3, 1), M(6, 1), N(3, 3) and its image
    after half turn about the origin.
   State the co-ordinates of the vertices L′, M′ and N′ of the image of a triangle.

(b) Rotate L′M′N′ of (a) above through +90o about the origin. State the
     co-ordinates of L′′, M′′, N′′.

12.4 Mixed composite  transformations in two dimensions

        Activity 12.5

Given triangle ABC has vertices
A(0, 2), B(2, 2) and C(2, 5).

(a) Plot triangle ABC on a Cartesian plane.

(b) Reflect triangle ABC in the line
     x = 3 to get the image A′,B′,C′.
Write the co-ordinates of A′B′C′.

(c) Translate triangle A′B′C′ bythe vectorto get the image
A′′B′′C′′. Write down the co-ordinates of A′′, B′′, and C′′.

An object can undergo several different transformations, one after the other.
This is done such that the image of the preceding transformation becomes the
object of the next transformation. When an object, A, undergoes a
transformation R followed by another transformation N, this is represented
as N[R (A)] or NR (A).
The first transformation is always indicated to the right of the second
transformation. RR (A) means ‘perform transformation
R on A and then perform R on the image’. It also written R2(A).

Example 12.9

A is a triangle with vertices (2, 1), (2, 4)
and (4, 1). R is a rotation of 90º. Clockwise about the origin and N is a
reflection in the y-axis. Find the vertices of the image of A if it undergoes the
following transformations:
(a) NR (A)                (b) RN (A)

Vertices of MRT (B) are (4,-4) (6,-4)(4,-7).

12.5 Inverse transformations in two dimensions

     Activity 12.5

The inverse of a transformation reverses the transformation, i.e. it is the
 transformation which maps the image back to the object.
The following are examples of inverse transformations are defined by the
 following cases.

1. If R denotes θº clockwise rotation, about (0, 0), then R^-1 denotes θo
    anticlockwise rotation about (0, 0).

2. If translation T has vector, the translation which has the opposite
    effect has vector. This is written as T^-1

3. For all reflections, the inverse is the same reflection. For example
    if X is reflection then X^-1 is also reflection.
    The symbol T^-3 means (T-¹)3 i.e. perform T^-1 three times.

Example 12.12

An image of a triangle with vertices A′(1, –2), B′(1,–4) and C′(6, –1) is
formed after an object ABC undergoes a clockwise rotation through 90º about the
origin. Find by construction the co-ordinates of its object, ABC.

Example 12.13

P′Q′R′S′ is the image trapezium whose vertices are P′(1,3), Q′(4,3), R′(1,1)
and S′(6,1). P′Q′R′S′ is obtained after reflection in
the x=0.
(a) Obtain the co-ordinates of object trapezium PQRS.

(b) Write down the co-ordinates of P′′Q′′R′′S′′ the image of PQRS
     when reflected in the line y=0.

Solution

All the co-ordinates can be obtained from graph 12.17 below.

• The line x=0 is y–axis.

• The line y=0 is x–axis.

Unit Summary

• In composite translation, all the points in the object move through the same
   distance and in the same direction

• Composite reflection: A reflection is a trans-formation representing a
   flip of a figure.
  A reflection maps every point of a figure to an image across a fixed line.
 The fixed line is called the line of reflection

• Equation of a mirror line: Equation of the mirror line can be
  found by using the co-ordinates of the midpoints of the image and the
  object.
  For example the object whose co-ordinates are A(a, b) and B(c, d),
  and the image A’(a1,b1), B’(c1,d1), The equation of the mirror line is
  got from the midpoints of A, A’ and B, B’ as;

and

The equation of the line joining points M1 and M2 is the equation of
the mirror line.
• Composite rotation: The turning of an object about a fixed point or
  axis is called rotation. The amount of turning is called the angle of rotation.

• Mixed transformations: An object can undergo several transformations,
  one after the other. This is done such that the image of the preceding
  transformation becomes the object of the next transformation.

• Inverse of a transformation: The inverse of a transformation reverses
  the transformation, i.e. it is the transformation which takes the image
   back to the object.

Unit 12 Test