Mathematics

Key unit competence

By the end of this unit, I will be able to transform shapes using congruence (central symmetry, reflection, translation and rotation).

Unit outline

• Definition of isometries i.e. central symmetry, reflection, translation and rotation.

• Construction of an image of an object/ geometric shapes under isometries.

• Properties and effects of isometries

• Composite transformations up to three isometries.

9.1 Introduction to isometries

Activity 9.1

1. Use a dictionary or internet to find the meaning of the following terms: 

(i) Transformation

(ii) Isometry

2. Compare and discuss your findings with those of other members of your class.

Using most geometrical shapes, we can transform a shape into a different shape or size or change position or direction etc. A figure or shape which has been altered in one way or the other is said to be transformed.  The procedure or process of alteration is called a transformation.

An isometry is that transformation that does not affect the size, the shape or area of the object being transformed.

The shape that is being transformed is called the object and the transformed figure, the image.

Referring to the object, the image and the transformation, we can say that an isometry maps the object onto the image. Some examples of common isometry are translation, reflection, rotation, and central symmetry.

In this chapter, we will deal with reflection, central symmetry, translation and rotation.

9.2 Central symmetry

9.2.1 Definition of central symmetry

Activity 9.2

1.  Copy figure 9.1, ABCDE in your note book and label the figure clearly as shown below.

                

2. Join point A to O. Extend line AO to A´ such that AO = OA´

3. Similarly join BO and extend it to B’ such that BO = OB´

4. Repeat procedure 1 for points C, D and E in order to locate points C´, D´ and E´, the images of C, D and E respectively.

5. Join the points A´B´C´D´E´ in that order to obtain a closed shape.

6. Describe figure A´B´C´D´E´ formed in relation to figure ABCDE.

7. How do the sizes, areas and shapes of ABCDE and A´B´C´D´E´ compare

From the activity 9.2,

• Figure ABCDE is identical to figure A´B´C´D´E´.

• Figure A´B´C´D´E´ is an inverted version of ABCDE.

• The two figures have same shape, same size and therefore same area.

• Activity 9.2 gives an example of central symmetry.

• Point O is the centre of the central symmetry.

Central symmetry is a transformation under which the image is inverted upside down about a point called the centre. The object and the image are equidistant from the centre, and the corresponding points lie on opposite sides of the centre.

9.2.2. Properties of central symmetry

1. An object and its image have same shape and size.

2. A point on the object and a corresponding point on the image are equidistant from the centre.

3. The image of the object is inverted.

4. Central symmetry is fully defined if the object and the centre are known.

Example 9.1

Triangle ABC has vertices at A(2, 1),  B(2, -4) and C(5, -4).

Find the image of ∆ABC under the central symmetry with centre O(0, 0). State the coordinates of the image

Solution

Let the image be A´B´C´ Fig 9.2 shows the object and its image. On graph paper plot the points A(2, 1) B(2, -4), C(5, -4) and mark the origin O(0, 0). Join the points A,B,C to form triangle ABC.

         

Join A to O and extend line to A´ so that OA´= OA and

mark point A´. Join B to O and extend line to B´

so that OB´= OB and point B´.

Similarly locate point C′so that     OC´= OC. 

The coordinates of the image ∆ are:   A´(-2, -1), B´(-2, 4), C´(-5, 4)

Exercise 9.1

1. Given that points A(4, 0), B(0, 3) and C(4, 3) are vertices of a triangle. Draw the triangle on a graph paper. Label this triangle clearly.Construct the image of ∆ABC under central symmetry, centre (0, 0).

2. Fig 9.3 below shows a triangle ABC and its image A´B´C´ under a certain transformation.

Identify the transformation and describe it fully.

3. Fig 9.4 below shows the image of figure ABCD under a central symmetry centre P.

(a) Copy the figure in your graph book and label it clearly.

(b) Accurately locate the object figure ABCD.

(c) State the coordinates of A,B,C and D.

4. Trace figure 9.5 below. Given that it represents an object and its image under a central symmetry, locate the centre of the transformations and label it O.

Use your construction to estimate the following:

(a) Lengths OA, OA´, OB, OB´, OC, OC´, OD, and OD´

(b) Angles between

  (i) OA and OA´ 

(ii) OB and OB´

(iii) OC and OC´

5. ∆ABC has vertices at A(1, 2) B(-2, 3) and C(-1, 5) the image of ∆ABC under central symmetry has vertices at A′(3, 2) B′(6, 1) and C′(6, -1). Using graph paper and a suitable scale, plot the points A, B, C, A′, B′ and C′

Join the appropriate points to show the distinct ∆s ABC and A′B′C. Use geometric construction to locate the centre, D, of the central symmetry. State the coordinates of D.

6. (a) Given that P′Q′R′S′ is the    image of PQRS under    a central symmetry,     and that P′(1, 1) Q′(6, 1) R′(5,   4) S′(2, 3) draw figure P′Q′R′S′   on a graph paper. If the centre of   central symmetry is C(4, -1), find   the coordinates of P,Q,R,S.

(b) If P′′Q′′R′′S′′ is the image of P′Q′R′S′ under a central symmetry, find the coordinates of the centre given that P′′(-3, -5), Q′′(-8, -5) R′′(-7, -8) S′′(-4, -7)

9.3 Reflection

9.3.1 Definition to reflection

We have already seen that the two parts of a shape on opposite sides of a line of symmetry, are mirror images of each other.

Activity 9.3

Look at yourself in a mirror. Do you  see yourself as others see you?  In what ways does your image differ from yourself?  Answer the following questions. 

1. If you raise your right arm, which arm appears to be is raised in your image? 

2. Which is taller, you or your image? 

3. If you stand 3 m infront of the mirror, where does your image appear to be in relation to the mirror? 

4. If you walk towards the mirror, what  happens to your image?

• Now consider figure 9.9 which shows Peter standing infront of a vertical mirror denoted by xy and his image on the other side of the mirror line.

• Draw a line joining the tip (N) of Peter’s nose to the tip (N’) of Peter’s image.

• Let the line NN´ meet the mirror line at a point O. Measure the distance NO, ON´ giving your answer to the nearest mm. What do you notice?

• Using a protractor, measure all the angles formed at point O. What do you notice?

From activity 9.3 you should have noticed the following

1. You and your image

• Are identical

• Face opposite directions.

• Stand same distance away from the mirror.

• If you walk towards the mirror your image walks towards you.

2. For Peter and his image

• NO´= NO´

• All the angles of point O are equal

i.e. each of the angles at O is 90°

Activity 9.3

above helps us to drive the properties of reflection as a transformation. Under reflection, the mirror is represented by a line called the mirror line.

9.3.2 Properties of reflection

The following are some of the properties of reflection as a transformation:

1. An object and its image have the same shape and size.

2. A point on the object and a corresponding point on the image are equidistant from the mirror line.

3. The image is laterally inverted, i.e. the object’s left-hand side becomes the image’s right-hand side and vice versa. Object and image face opposite directions. They are oppositely congruent.

4. The line joining a point and its image is perpendicular to the mirror line.

5.  A point on the mirror line is an image of itself.  Such a point is said to be invariant since its position does not change.

6. A reflection is fully defined if an object point and its image are known or one point and the mirror line one given.

Note: We think of a mirror as two-sided so that if object point B is on the same side as the image A′, then its image B′ is on the same side as the object A  (Fig. 9.7)

                 

Example 9.2
 Draw the image of triangle PQR (Fig. 9.8) under reflection in the mirror line m.

            

Solution

(i) To obtain the image of point P, use a pair of compasses and a ruler to draw a perpendicular from P to the mirror line and produce it (Fig. 9.9).

(ii) Mark off P′, the image of P, equidistant from the mirror line as P.

(iii)Similarly, obtain Q′ and R′, the images of Q and R respectively in the same way.

(iv)Join P′, Q′ and R′ to obtain the image of  ∆PQR.

Exercise 9.2

1. Trace each of the drawings in Fig. 9.10 below and construct their images under reflection in the indicated mirror line m.

2. Fig 9.11 shows objects and their images under reflection. Trace each of the drawings and construct the mirror line in each case.

9.3.3 Reflection on the Cartesian plane

Reflection in the mirror lines x-axis
 (y = 0) and y-axis (x = 0)

Activity 9.4
• Consider a triangle of sides with   the coordinates A(1, 4), B(3, 5) and   C(4, 12).

• In the cartesian plane provided,   plot the points A, B C and join the   points.

• Identify the x and y axes and reflect   the triangle along these axes.

• What do you notice about the    shape and the distance of the image   to the object from the line of    reflection?

• Discuss with the whole class

Discussion

x and y - axes are lines y=0 and x=0 respectively. Reflection along these axes takes the object's image to the opposite side of the axis with both the object and image distances from the axis equal. The image remains the same as the object.

Example 9.3

A(2 ,4 ),  B(6, 4)  and  C(7 , 2) are the vertices of a triangle. Find the image of the triangle under reflection in the line  (i) x-axis, (ii) y-axis, labelling them as A′B′C′ and A′′B′′C ′′ respectively.

Solution

Fig. 9.12 shows ∆ABC and its images.

• x-axis is the line y = 0 while y –axis is the line x = 0.

•  To construct the images, a perpendicular line is dropped to these lines y = 0 and x = 0 respectively.

From example 9.3 above, how are the x-coordinates and y-coordinates of an object point and its image related?  How are the y-coordinates related? From the example, you should be able to notice that

9.3.2.1 Reflection in the mirror lines   x = k and  y = k

Example 9.4

Find the images of ∆ABC with vertices A(–1 , –2), B(1 , 5)  and  C(2 , 3) under reflection in the mirror lines  (i) x = –1,  and  (ii) y = 1, labelling them as ∆A′B′C′ and ∆A′′B′′C′′ respectively.

Solution

First, you need to identify the lines x = –1 and y = 1 on the Cartesian plane. Then we obtain the image by first reflecting it on the line x = –1 followed by a reflection on y = 1. ∆ABC and its images are shown in
 Fig. 9.13.

What do you notice about the relationship between the x-coordinates and y – coordinates of an object point and its image from example 9.3 above?

You should notice that:

9.3.2.2 Reflection in the mirror lines y = x and y = –x

Example 9.5

A(–1,2), B(1,5) and C(3,4) are the vertices of a triangle. Find the images of the triangle when it is reflected in the mirror lines (i) y = x, and (ii) y = –x, labelling them as A′B′C′ and A′′B′′C′′ respectively.

Solution

Fig. 9.14 shows ∆ABC and its images. The same method of constructing the image is applied here where the two lines y = –x and y = x are first identified on the plane then a perpendicular is constructed.

For example 9.5 above, how are the x-coordinates and y-co-ordinates of an object point and its image related?

You should notice that:

Example 9.6

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 then reflect the image in the line y = –x.

Solution

We first obtain the image under reflection in line y = 0. Then we reflect this image in line y = –x. This is shown in Fig. 9.15  In the figure, 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"(0.5 , –2), B"(2 , –2), C"(3.5 , –4) and D"(1 , 3.5)

Exercise 9.3

1. A quadrilateral has vertices P(4 , 2),Q(7 , 3),R(6 , 2)  and  S(4 , 0).  Draw, on the same axes, the quadrilateral and its images under reflection in

(a) the x-axis,

(b)   the line y = x,

(c) the y-axis,

(d)   the line y = –x, labels the images as P′Q′R′S′,  P′′Q′′R′′S′′, P′′′Q′′′R′′′S′′′ and  P′′′′Q′′′′R′′′′S′′′′ respectively .

State the coordinates of each image point.

2. The vertices of a triangle are A(–4 , 6),  B(–3 , 2)  and  C(–7, 1).  Find the final image of the triangle under

(a)(i)  Reflection in line y = 0

(ii) Reflection of the image in (i) In line y = x.

(b) (i)  Reflection in line y = –x

(ii)  Reflection of the image in (i) In line x = 0.

(c) (i)  Reflection in line y = x

(ii) Reflection of the image in (i) In line y = 1.

(d) (i)   Reflection in line x = 1.5

(ii)  Reflection of the image in (i)   In the same line.

3. Under reflection, which properties of an object are invariant?

4. ∆ PQR has vertices P (2, 1), Q (4, 3) and  R (3, 5) and those of ∆ P′Q′R′ are P′ (2, –1), Q′ (4, –3) and R’ (3, –5). On the same axes, draw the two triangles. Describe the transformation that maps ∆ PQR onto ∆P′Q′R′.

9.4 Rotation

9.4.1 Introdcution to rotation

Rotation is another example of an isometric transformation.

Activity  9.5

Draw a triangle OAB, as shown in Fig. 9.16. Trace the triangle using tracing paper.  Name the vertices of the tracing O′, A′ and B′ to correspond with O, A and B.

               

Place the tracing exactly on top of the original figure. Put a pin through O and O′. Keeping the lower sheet still, rotate the tracing anti-clockwise about O through an angle of approximately 90°.  Answer the following questions.

(a) Through what angle has each of lines OA and OB turned?

(b) Which is longer, AB or A′B′; or they are of the same length?

(c) Have any points remained in the same position?  If so, which ones?

(d) Are the angles of the image the same as the corresponding angles of the object triangle?

Put the tracing back exactly on top of the original triangle.  Rotate the tracing clockwise about O through 90°.  Answer the above questions now.  Do you get the same result as before?

The point about which a figure is rotated is called the centre of rotation and the angle through  which the figure is rotated is called the angle of rotation.

9.4.2 Direction of rotation

Activity  9.6

• Using the same figure and tracing as in Activity 9.5, arrange the tracing to coincide with the original triangle again.

• Rotate the tracing about O through 60°.  How do you know when to stop rotating the tracing?

• You may do this by first marking the final position of OA on the lower sheet before putting the tracing on top.  Fig 9.17 shows the lower sheet with the image position of OA drawn as broken line OA′. OA′ is called a guide line.

• Arrange the tracing to coincide with the original triangle. 

• Rotate the tracing about O until OA′, on the tracing, comes on top of the guide line.

         

• Is OB turned the same angle as OA? Measure to check your answer.

• Is any point remained fixed in this rotation?

• What is the angle between AB  and A′B′? (Extend the line segments if necessary).

The direction of rotation is important! An anticlockwise turn is referred to as positive turn while a clockwise turn is referred to as negative turn. Therefore, an anticlockwise turn of 90° is called a rotation of +90° while a clockwise turn of 90° is called a rotation of –90°

Note:  This convention is used throughout in mathematics, science and engineering.  It is only in the measurement of bearings that the positive direction is clockwise.

Activity  9.7

• Draw another triangle as in Activity 9.6 and label its vertices A, B and C.

• Mark on the lower sheet a point O, which is not on the triangle.

• Rotate the tracing about O through an angle of –90°.  How do you do this?

• You may do this by using guide lines.

• In Fig. 9.18, OD has been drawn. This will be rotated to the position OD′

       

• On your lower sheet, mark the guide lines OD and OD′.

• Place the tracing back on the figure and trace OD.

• Now rotate the tracing through –90° about O.

• How do you know when to stop rotating the tracing?

• What size is the angle between a line and its image in this rotation?  (Extend the line segments, if necessary, to answer this).

• What conclusion can you draw?

• Which is longer, OA  or its image OA′?  What about OB and OB′, OC and OC′?

• Instead of rotating through –90°, through what angle would you have to rotate the tracing in positive direction to get into the same position?

• We say that a rotation of 270° has the same effect as one of –90°.

• What positive rotation has the same effect as one of –150°?

• What negative rotation is equivalent to a rotation of 320°?

9.4.3 Properties of rotation

From Activities 9.5 to 9.7, you should have noticed that rotation has the following properties:

1. All points on the object turn through the same angle in the same direction.

2. The angle between a line and its image equals the angle of rotation.

3. Each point and its image are the same distance from the centre of rotation.

4. The centre  of rotation is invariant i.e. it does not change its position.

5. The object figure and its image are identical i.e. directly congruent.

6. A rotation is fully defined when the centre and direction angle of rotation are specified.

7. A positive rotation through an angle θ is the same as a negative rotation through an angle of  (360°–θ) about the same centre.

8. Rotation preserves shape and size.

9. By convention, a clockwise rotation is negative and anticlockwise rotation is positive.

9.4.4 Rotation and congruence

Activity 9.8

Refer back to Activities 9.5 to 9.7 you did before.

Looking at the object and the image. What can you say about;

(a) the sizes of corresponding angles?

(b) the lengths of corresponding sides?

(c) the orientation (i.e. the direction in which they face)?

Under rotation, an object and its image are directly congruent.

9.4.5  Locating an image given the object, centre and angle of rotation

Activity 9.9

• Identify point M, the midpoint of line segment AB and join it to point C.

• Identify point N, the midpoint of line segment AC and join it to point B.

• Let CM and BN meet at a point O.

• Using point O as the centre and 180° as the angle of rotation, draw the image of ∆ ABC under the rotation. Label the image as ∆ A´B´C´.

• Describe the congruency of the two triangles.

This activity helps you to find the image of a given figure provided the centre and angle of rotation are known. In this case, O is the centre of rotation and 180° is the angle of rotation

Example 9.7

Fig. 9.20 shows a triangle PQR in which  PQ = 3 cm, QR = 4 cm and PR = 5 cm. Copy the figure and locate ∆P′Q′R′, the image of ∆PQR, under a  rotation of 65° about point O.

Solution

To locate ∆P′Q′R′, proceed as follows:

(a) Join P to O. W ith OP as the initial line,  measure an angle of 65° anticlockwise at O and draw a construction line OA. (Fig. 9.21).

(b) To obtain P ′ on OA, measure OP ′ = OP.  Mark the point P′.

(c) Repeat step (a) for Q and R to obtain  construction lines OB and OC respectively. Measure OQ′ = OQ and OR′ = OR on OB and OC to obtain points Q′ and R′.

(d) Join P′, Q′, R′ to obtain ∆P′Q′R′

9.4.6 Finding the centre and angle of rotation

Activity 9.10

• Trace or copy Fig. 9.22 below in your exercise book

• To join P to B′, C to C′ correct the OA and OA′ include O′

• Construct perpendicular bisectors of BB´ and CC´ to meet at point O.

• Join (i) OA and OA′, (ii) OB and OB′ and (iii) OC and OC´. What do you say about the lengths of the pairs of line segments?

• Measure angles (i) <AOA´ (ii) <BOB´ (iii) <COC´ and comment about your answer.

• Describe the significance of the point O with reference to the triangles ABC and A´B´C´.

• Describe the meaning of the size of the angels; AOA´, BOB´ and COC´

Note

A rotation is fully defined if both the object and the image are known. Under rotation, every point of an object moves along an arc of a circle whose centre is the centre of rotation.  Thus, if a point A is mapped onto a point A′ by a rotation about a point O, then AA′ is a chord of the circle centre O, through A and A′ (Fig. 9.23).

                 

The perpendicular bisector (mediator) of a chord of a circle passes through the centre of the circle.

Thus, the perpendicular bisector (mediator) of AA′ passes through the centre of rotation O. We use this fact in locating the centre of rotation.

Example 9.8

In Fig. 9.24, ∆A′B′C′ is the image of ∆ABC after a rotation. Copy the figure and locate the centre of rotation. Determine the centre and angle of rotation

                   

Solution

To locate the centre of rotation, proceed as follows:

(a) Join A to A′ and construct the mediator of  AA′ (Fig 9.25).

(b) Join B to B′ and construct the mediator of BB′.

(c) Produce the mediators in steps (a) and (b) so that they intersect at point O.

(d) Construct the mediator of CC′. This mediator also pass through O. Using the method of Activities 9.9 to 9.11, check that O is actually the centre of rotation.

                     

To find the angle of rotation, join any one of the points A, B or C to the centre of rotation O. Also join the corresponding image point to O. Measure the angle thus formed. The angle of rotation is –120°.

To find the centre of rotation, we draw the mediators of the line segments formed by joining object points to their corresponding image points.  As all the mediators pass through the centre of rotation, it is sufficient to find the intersection of any two mediators. To find the angle of rotation, we join a pair of corresponding points to the centre of rotation, then measure the angle formed at the centre and specify the direction of the rotation.

Exercise 9.4

1.The drawing in Fig. 9.26 has a rotational symmetry of order 2

                                

(a) What point is the centre of rotational symmetry?

(b) Which lines are parallel?

(c) What is the image of point S?

(d) If the points P, Q, T, and S are joined, what kind of quadrilateral is formed?

(e) If PT = 9 cm, what is the length of RT? (f) If ∠RST = 48°, what other angle is 48°?     

2. PQ is a chord of a circle centre O (Fig. 9.27). The circle is rotated about O so that P′Q′ is the image of PQ.

               

(a) What can you say about the length of  PQ and P′Q′?

(b) What is the perpendicular distance of P′Q′ from O, given that PQ is a perpendicular distance x from O?

(c) Copy and complete the following statement: Equal chords of a circle are the same ______ from the centre of the circle.

3. Each part of Fig. 9.28 shows an object and its image after rotation

(a) Trace the diagrams and find the centres of rotation.

(b) Find the angle of rotation in each case giving your answer both as a positive and a negative angle.

              

9.4.7 Rotation in the cartesian plane

Activity 9.1

Consider Fig. 9.29 on the cartesian plane, showing a triangle ABC and its images after rotations with different angles of rotation.

1. With ∆ABC as the object and ∆A′B′C′ as the image:  

(i) What is the centre of rotation? 

(ii) What is the angle of rotation?

(iii)  Copy and complete Table 9.1.

2. With ∆ABC as the object and ∆A′′B′′C′′ as the image:

(a)   What is the centre of rotation?

(b)   What is the angle of rotation?

(c)    Copy and complete Table 9.2

3. With ∆ABC as the object and ∆A′′′B′′′C′′′ as the image;

(a)   What is the centre of rotation?

(b)   What is the angle of rotation?

(c)   Copy and complete Table 9.3.

From activity 9.11 you should have noticed that

A rotation about the origin (0, 0);

1. through 90° maps a point (a, b) onto the point (–b, a).

2. through –90° maps a point (a, b) onto the point (b, –a).

3. through 180° maps a point (a, b) onto the point (–a, –b).

Where do rotations of 0° and 360° about the origin map point (a, b)?

Activity 9.12

Consider a quadrilateral with the vertices A(2, 5),  B(4, 5),  C(6, 3) and D(3, 2). With (1, 2) as the centre of rotation, rotate quadrilateral ABCD through 180°.

From Activity 9.12, you should notice that a rotation of 180° about a point (1, 2) maps a point (a, b) onto the point (2 × 1 – a,  2 × 2 – b).

A rotation of 180° about (h, k) maps a point (a, b) onto the point   (2h – a, 2k – b).

Example 9.9

ABCD is a rectangle whose vertices are A (1, 0), B (4, 0), C (4, 2) and D (1, 2). Find the co–ordinates of the image of the rectangle A´∴B´∴C´∴D´∴, of ABCD after a rotation of –90° about (0, 0)

Solution

On a graph paper or a squared paper, draw rectangle ABCD. Since the rotation is negative, it is in a clockwise direction, measure AOA′= 90º to locate A′ the image of A, such that OA = OA′. Similarly, measure ∠BOB′ ∠COC′ = ∠BOB′ = 90º and BO = OB′, OC = OC’, OD = OD′to locate B′,C′and D′, the images of B,C, and D respectively. Fig. 9.30 below shows both the object and its image rectangle.

The co-ordinates of the image are: A′(0,−1), B′(0,−4), C′(2,−4) and D′(2,−1)

Example 9.10

A triangle with vertices A (1, 3), B (2, 1) and C (3, 1) is mapped onto another triangle with vertices A′ (-3, 1), B′ (-1, 2) and C′ (-1, 3). Describe this transformation fully.

Solution

The object and its image are shown in Fig. 9.31.

Both the object and its image are directly congruent. Since ∆ ABC cannot map onto         ∆ A′B′C′ by a reflection or a translation, the transformation must be a rotation. To find the centre of rotation;

(i) Join AA′ and construct its perpendicular bisector. (ii) Join BB′ and CC′ and construct the perpendicular bisectors. The perpendicular bisectors meet at the centre of rotation referred to as point O. Hence, the centre of rotation is (0, 0) i.e. about the origin.

To find the angle of rotation;

(i) Join AO and A′O

(ii) Measure ∠ AOA′.           

∠ AOA′ = 90º

So, a rotation about point O maps OA onto OA′. Similarly, OB → OB′ and OC → OC′.

Since ∠AOA′ = 90º,

∴ the transformation is a rotation centre (0,0) angle 90º. This is a positive rotation.

Note: The angle of rotation can also be given as –270º also written as 270º; clockwise.

Exercise 9.5

1. L(4, 2),  M(–1, –2) and N(3, 0) are the vertices of a triangle.  Plot these points on a squared paper and with C(2, 1) as the centre, rotate LMN through an angle of 90°.

(a) Write down the coordinates of L′, M′ and N′.

(b) If S is the point (2, –1), what are the coordinates of S′?

(c) If T is the point (3, 4), what are the coordinates of T′?

(d) Without measuring, state the angle between LM and L′M′.

(e) What is the path traced out by L in moving to L′?

2. A(–3, 1),  B(1, 1),  C(1, –3),  D(–3, –3) and P(–1, 3), Q(3, 3),  R(3, –1),  S(–1, –1) are the vertices of two squares ABCD and PQRS. Describe fully the rotation that maps:

(a) ABCD onto QRSP (this means that A is mapped onto Q, B onto R, and so on).

(b) ABCD onto SPQR, and

(c) ABCD onto RSPQ.

3. Write down the images of the following points under rotation through the given angles and about the stated centres.

(a) Centre (0, 0), angle 90º 

(i) (4, 4)

(ii) (3, –4) 

(iii) (4, –7)

(iv) (–6, –8)

(b) Centre (0, 0) angle –90º

(i) (7, 1)

(ii) (–3, 6)

(iii) (4, –7)

(iv) (–2, –3)

(c) Centre (0, 0), angle 180º: 

(i) (4, 4)

(ii) (–3, 2)

(iii) (0, –5)

(iv) (–3, –4)

(d) Centre (3, 2) angle 180º: 

(i) (2, 3)

(ii) (–5, 3)

(iii) (4, –5)

(iv) (3, –1)

(e) Centre (–2, –5) angle 180º:

(i) (7, 1)

(ii) (–3, 6) 

(iii) (4, –7)

(iv) (–2, –3) 

 4. A negative quarter turn about the point (0, –1) maps ABC onto A′B′C′ with the vertices A′(3, 1), B′(0, 5), and C′(0, 1). Find the vertices of ABC.

5. A quadrilateral has vertices A(1, 3) B(2, 5) C(4, 4) and D (3, 3).

(a) Find the coordinates of the image quadrilateral A′B′C′D′ under reflection in the line y = 2.

(b) A certain transformation maps points A′, B′, C′ and D′ onto points A′′(–5, –3),  B′′(–7, –3), C′′(–8, –2), and D′′(–6, –1) respectively.  Describe this transformation fully.

6. ∆ LMN has vertices L (2,3), M (2,5),     N (6,5). Find the coordinates of L′M′ and  N′  under the following transformations: 

(a) Rotation of 90o about (0, 0)

(b) Rotation of –90o about (0, 0)

(c) Rotation of 180o about (0, 0)

7. A triangle has vertices at X (3, 5), Y (3, 2) and Z (5, 2). Describe fully the transformation that maps:

(a) ∆ XYZ onto ∆ X′Y′Z′ whose vertices are X′ (–3, –5), Y′ (–3, –2) and  Z′ (–5, –2).

(b) ∆ X′Y′Z′ onto ∆ XYZ given that  vertices of X′Y′Z′ are X′ (–3, 5), Y′ (–3, 2) and Z′ (–5, 2).

 9.5 Translation

9.5.1 Definition of translation

Activity 9.13

2. Draw the line segment joining B to B′ as shown in Fig. 9.37 (b).

3. Slide the tracing using line BB′ as a guide line, to ensure that B moves onto B′ in a straight line.

4. When B coincides with B′, stop the slide. What do you notice about the positions of A and C?

5. What do you notice about the new position of ∆ABC?  What can you say about the two triangles?

From Activity 9.13, we notice that each point on triangle ABC has moved the same distance in the same direction. The process that moves triangle ABC onto triangle A′B′C′ is called a translation.

Note:

In our previous work, under properties of reflection and properties of rotation we noticed that under reflection and rotation, sides, angles and area were invariant. This is similar to the findings we have observed under translation. Thus, a translation is described using distance and direction.

Properties of a translation

Translation encompasses the following properties:

1. All the points on the object move the same distance.

2. All the points move in the same direction.

3. The object and the image are identical and they face the same direction. Hence, they are directly congruent. This means that shape, size, angles and area are invariant.

4. A translation is fully defined by stating the distance and direction that each point moves.

Activity 9.14

From Activity, 9.16, you will notice that the translation that maps A onto A′ is defined by the distance |AA₁| and the direction AA₁. Similarly, the translation that maps A onto A₂ can be defined by the distance|AA₂| and the direction AA₂. The displacement vector in any of the translations can be given by any BB₁, CC₁, DD₁ or BB₂, CC₂, DD₂ but the distance moved remains the same.

All the displacement vectors that describe a translation must be equal and therefore parallel, i.e. AA₁, BB₁, CC₁, DD₂ are equal. Similarly, AA₂, BB₂, CC₂ DD₁ are equal and therefore parallel.

Exercise 9.6

1. A packaging case is pushed (without turning) a distance of 8 m, in a straight line (Fig. 9.34).  The corner of the box which was originally at A ends up at B.

How far will each of the following have moved?

(a) The upper front edge.

(b) Each vertex.

(c) The centre of each face.

(d) The centre of the box.

Fig. 9.35 shows a tiling composed of congruent parallelograms of sides 10 cm by 5 cm. Suppose the lines in the diagram are used as guide lines along which tiles may be slid.

2. (a) If tile 6 moves onto tile 7, in what direction has each vertex of the tile moved? How far has the point Xmoved?  Make a statement that is true for every point on tile 6.

(b) Tile 6 moves along the guideline parallel to line BC in th direction BC. Observe and state the direction of motion of:

(i)  each vertex of tile 6.  

(ii)   the point X on tile 6. 

(iii)   every point on tile 6.

(c) Answer Question (b) for the case where tile 6 is slid to position 14.

3. Name the tiles onto which tiles 1, 11, 15 and 18 will be translated by a translation equivalent to that of Question 2 (a).

4. (a) Name the tiles onto which tiles,  2, 8, 11  and 5 will be translated by a translation equivalent to
  that of Question 2 (c).

(b) What will be the images of letters E, F, G and H under the same translation in 2 (c) above?

5. Write down all the possible translations that are equal to the translation:

(a) FH

(b) AC

9.5.2 Translation in the Cartesian plane

Activity 9.1

Consider Fig 9.36 below. ∆ PQR is the image ∆ ABC under a translation. Described by the vector AP, BQ, CR or any other vector equal to the column vector. What do you notice about the translation shown

In this case, vector (4 2) defines the translation that maps ABC onto PQR. When using the Cartesian plane, a translation is fully defined by stating the distances moved in the x and y directions.  The column vector that  defines the translation is also called the displacement vector of the translation.

On the Cartesian plane, if a point P (x, y) is mapped onto anther point P′ (x + a,  y + b), we say that P is mapped onto P′ by the translation (a b). The column vector   ( a b)  defines both the distance and the direction, where a represents the horizontal distance moved and b the vertical distance moved

In general,

A displacement vector AA′ is a quantity which describes a change in position from a point A to A′.  It can be defined by giving the length AA′ and  the direction of A′ from A or by giving a displacement vector called a column vector. This means that a translation can be performed if:

(i) the object and the displacement vector are known or

(ii) the object and the image of one (1)  point are given or

(iii) the displacement vector and the image are given, to find the object.

Example 9.11

Triangle ABC has vertices A (0, 0),   B (5, 1) and C ( 1, 3). Find the coordinates of the points A′, B' and C', the images of A, B and C respectively, under a translation with displacement vector (2 5  ) .

Example 9.12

Under a translation that maps a point P (2, 3) onto P ′(4, 7), rectangle ABCD is mapped onto another rectangle A′B′C′D′. Given that the vertices of A, B, C and D are (1, 4), (5, 4), (5, 2) and (1, 2) respectively, perform the following: (a) calculate the co–ordinates of   A′B′C′ and D′. (b) describe the translation that would map A′B′C′D′ onto ABCD. (c) On the same axes, represent both the object  and the image rectangles, and use them to verify your answers to (a) and (b) above.

Example 9.13

Triangle A′B′C′ is the image of ∆ABC under a translation. Given A′(0,  –3)  B′ (1, –5),  C′(1, –2);  A, B and C are points (–2, –2),  (–1, –4), (–1, –1) respectively, find the translation vector.

Note that the result of the combined translation is the same as that obtained by doing the individual translations one after the other.

Exercise 9.7

1. Quadrilateral A′  B′ C′ D′ is the image of ABCD under a translation. Given that the vertices A, B, C and D have co-ordinates (2, 2), (5, 2), (5, 5) and (2, 5) respectively and that A′ is the point (9, 2) find:

(a) the translation vector

(b) the co-ordinates of: 

(i)   B' 

(ii)  C´

(iii)  D´

2. Given the rectangle ABCD in question 1 and a displacement vector 6 3 , find the co-ordinates of the image of ABCD under this translation.

3. A triangle has vertices at L (1, 1); M (6, 1) and N (5, 4). Another triangle has vertices at L′ (3, 4), M′ (8,4) and N′ (7, 7).

A student in your class claims that ∆ L′M′N′ is the image of ∆LMN under a translation. Do you agree? Explain your answer.

4. The position of a point P in a Cartesian plane is described as P (x, y). Describe the transformation, that maps point P (x, y) onto the point

P′ (x + 5,  y + 2).

5. (a) The points A and B have  co–ordinates (3, –1) and (3, 1) respectively. Given  that O is the origin, state the column vectors for the translations:

  (i) OA    

(ii) AB   

(iii) OB.

(b) Find the co-ordinates of O′, A′ and B′ under each of the translations in (a).

6. A square has vertices at A (0, 1)  B(2, 1), C(2, 3) and D(0, 3). Its image under a translation has vertices at  A′(5, 5), B′(7, 5), C′ (7, 7) and D′(5, 7). Describe the following transformations:

(a) the one that maps ABCD onto A′B′C′D′.

(b) the one that maps A′B′C′D′ onto ABCD.

7. A certain translation maps P (3, 5) onto P′ (7, 8). Find the co–ordinates of the point which is mapped onto Q′ (2, 7) under the same translation.

8. A quadrilateral P′Q′R′S′ is the image of PQRS under a translation. Given vertices P (2, 0), Q′ (0, 3), P′ (5, 4), R′ (6, 9) and S′ (9, 4), find:

(a) the translation vector.

(b) the co-ordinates of R, S and Q.

(c) the translation that maps P′Q′R′S′ onto PQRS.

9.6 Composite Transformations

It is possible to combine more than one transformation using the same object. In such a case we say we are working with composite transformation.

Using the same diagram

• Using graph paper, draw OABC such that O(0, 0) A (1, 0) B(1, 1) and C(0,1)

• Find the image O′A′B′C′ of OABC after a reflection in the x-axis.

• Find the image O′′A′′B′′C′′ of O′A′B′C′ after a rotation of 90° about the origin.

• Now, translate O′′A′′B′′C′′ to O′′′A′′′B′′′C′′′ using displacement or translation vector  1 2 .

• State the coordinates of each image and ensure that the plotting is accurate.

• Discuss your findings with other members of your class.

From activity 9.16 above, the final image is identical to the original object to mean properties of isommetries are not altered by the combination of the transformations.

In this example, we did a reflection in the x-axis, followed by a rotation centre (0, 0) angle 90° followed by a translation with vector  (1 2) In this case order does not matter.

Properties of composite transformations

1. If  X and Q represent two transformations, then  XQ meansperform transformation  first followed by transformations X. i.e. if A represents an object,  XQ(A),  Q(A) = A′ (the image of A under Q  and XQ (A) =  X(A′) = A′′

2. Similarly if XQ and R are three transformations, and B is an object,  XQR(B) = XQ(B′) (B′ image of B        under R)      =  X(B′′) (B′′ is the         image of B' under Q)        = B′′′ (the image of B′′       under X)

3. If X represents a reflection under a certain mirror line, XX means a reflection in a line followed by a reflection in the same line i.e. XX(A) = X(A′)        

  = A′′ This means A′ = A′′

Note:   XX can be written as X2

4. A translation followed by a translation equals another translation.

5. A rotation followed by another rotation about the same centre results in another rotation.

6. In general for two transformations  X and Q, QX ≠ XQ.   

7. Composite transformations are performed successively in the given order as in point 2 above.

Example 9.16

ABC is a triangle with vertices A(1, 2), B(3, 1) and C(2, 3) Find the coordinates of:

a) (i)  the image of ∆ABC after    a reflection in the x-axis. Let   the image ∆ have vertices A´,   B´ and C´.

(ii) the image of A´B´C´after a   rotation of 180° about (4,0).   Let the new image have vertices   A′′, B′′ and C′′

(iii)the image of ∆A′′B′′C′′ after a   central symmetry centre (7, 3).

b) Describe the transformation that would map

(i) ∆ABC onto ∆A′′B′′C′′

(ii) ∆A′B′C′ onto ∆A′′B′′C′′

Solution

On graph paper, plot and draw ∆ABC and label it clearly. On the same diagram draw the three images and label them appropriately.  a)  Fig 9.39 shows the three images of ∆ABC

(i) Image of ABC under reflection is divided as ∆A′B′C′ with vertices A′(1, -2) B′(3, -1) C′(2, -3)

(ii) Image of ∆A′B′C′is directed as ∆A′′B′′C′′with vertices A′′(7, 2) B′′(5, 2) C′′(6, 3)

(iii) Image of ∆A′′B′′C′′ is denoted as ∆A′′B′′C′′with vertices A′′(7, 4) B′′(9, 5) C′′(8, 3).

  b)(i) By observation, ∆A′′B′′C′′ is the   image of ∆ABC under a reflection.   By  construction the mirror line   passes through the perpendicular   bisect  of AA′′or BB′′or CC′′. The   line has the equation x = 4

(ii) ∆ABC  maps onto ∆A′′B′′C′′ by a reflection in the line x = 4

(iii) ∆A′B′C′and A′′B′′C′′ are identical and face the same direction ∆A′B′C′maps onto ∆A′′B′′C′′by a translation, vector (6  6 )

Exercise 9.8

1. Figure ABCD has vertices at 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° about the origin.

(ii) Draw the image of A′′B′′C′′D′′ of A′B′C′D′ under a reflection in the line y = x. State the coordinates of A′′, B′′, C′′ and D′′

(b) A′′′B′′′C′′′D′′′ is the image of A′′B′′C′′D′′ under reflection in the line y = 0. Draw figure A′′B′′C′′D′′ and state its co-ordinates.

(c) Describe fully the transformation that maps

(i) ABCD onto A′′′B′′′C′′′D′′′

(ii) ABCD onto A′′B′′C′′D′′

2. ∆ABC has vertices A(1, 1), B(1, 3) C(4, 3) ∆A′B′C′ is the image of ∆ABC under a certain transformation P, so that A′(-1, 1), B ′(-3, 1) and C′(-3, 4) ∆A′′B′′C′′ is the image of ∆A′B′C′under another transformation, M. The vertices of ∆A′′B′′C′′ are   A′′(-1, -1) B′′(-3, -1) C′′(-3, -4)

(a) On the same diagram draw ∆ABC and its two images.

(b) Describe the transformation denoted as P.

(c) Describe the transformation denoted as M. (d) Describe a single transformation that maps ∆ABC onto ∆A′′B′′C′′

3. M is a reflection in the line y = x.  T is a translation that maps the origin  (0, 0) onto the point (10, 2). Given that ∆ABC has vertices at A(-2, 6), B(2, 3) C(-2, 3)

(a) Find the coordinates of (i) M(A) (ii) T(B) (iii) TM(A) (iv) MT(B)

(b) Find the image of ∆ABC under a combined transformation (i) TM (ii) MT State the coordinates b(i) and (ii)

4. M is a reflection in the line y = -x. H is a central symmetry centre (0, 0). The vertices of triangle P are A(0, 1), B(0, 6) C(4, 6). 

Find the coordinates of P′ under the following transformations

(a) M(P)

(b) H(P)

(c) HM(P)

(d) MH(P)

(e) MM(P)

(f) HH(P)

(g) Comment on the results of the transformations in parts (c) to (f).

Unit summary

1.  Isometry: this is a transformation which preserves shapes, appearance, size and area of the object. Examples of isometries are

• Central symmetry

• Reflection

• Rotation

• Translation

2. Central symmetry: is fully defined if the object and the centre are known

• Object and image are identical, but are inverted

• ∆A’B’C’ is the image of ∆ABC by a rotation, 180° about the centre O.

• Object point, corresponding image point and centre are collinear.

3. Rotation: is defined [if given one point on the object and the centre and angle of rotation or

• Two point and their corresponding images.

• An angle of rotation can be stated as positive (anticlockwise) or negative (clockwise).

• Both object and image are described as being directly congruent

4. Reflection

• We define a reflection by giving a point on the object and the mirror line.

• Object and image are identical but face opposite directions. They are said to be oppositely congruent.

• Corresponding points under reflection are equidistant from the mirror line and the line segment joining them meets the mirror line at 90°

• Points on the mirror line are invarian.

5. Translation

A translation is fully defined if

• If the object and the translation vector is given or An object and corresponding image point are known

• Under translation, points move equal distance in the same direction i.e.  parallel.

• Both object and its image are identical and they are said to be directly congruent. They face the same direction.

6. Composite transformations

These are transformations that are performed successively on the same object. They are also known as combined transformations.

Unit 9 test

1. Copy paste each shape and on the copy, draw a line of symmetry.

2. Draw a line segment PQ on a piece of paper. Does PQ have a line of symmetry? Fold the paper so that the fold is a line of symmetry of  PQ.  What is the size of the angles between the fold and PQ?  What can you say about the distances of P and Q from any point on the line of symmetry?

3. A(–4 , 1),  B(–2 , –1),  C(1, 0) are the vertices of a triangle.  Find the image of the triangle when it is reflected in the mirror line:

(a) y =  1     

(b)  y = –2

(c) x = –3     

(d)  x = 1.5

4. ∆A′B′C′ has vertices A′(–2, –1),   B′(–2, –4) and C′(–4, –4). Find the  co–ordinates of  the vertices of ∆ABC such that ∆A′B′C′ is the image of ∆ ABC under half turn about the origin.

5. ∆ ABC is mapped onto ∆ A′B′C′ under a given transformation. Given  that the co–ordinates of ∆ ABC are A (–3, 4), B(–3, 1), C (–1, 1) and those of ∆A′B′C′ are A (–3, –4),  B′ (–3, –1) and  C′ (–1, –1), describe fully the given transformation. 6. A triangle has vertices A (1, 2), B (7, 2) and C (5, 4)

(a) Draw triangle ABC on the Cartesian plane.

(b) Construct  ∆ PQR, the image of ∆ ABC under a rotation of 90o clockwise about the origin.

(c) On the same axes, draw ∆ XYZ; the  image of ∆ PQR under a reflecion in the line y = x. State the co–ordinates of x, y and z.

(d) ∆ MNS is the image of ∆ XYZ under a reflection in the line  y = 0.State the co–ordinates of M, N and S.

(e) Describe one transformation that maps ∆ MNS onto ∆ ABC.

7. A square PQRS has vertices at P (2, 2), Q (2, 6), R (6, 6) and S (6, 2). Find the co–ordinates of the image of the square under a reflection in the line y = x.

8. On the same axes, draw ∆ ABC and ∆A′B′C′. Given that the co–ordinates of A, B, C, A′, B′ and C′ are (3, 4),  (7, 4),  (7, 6), (4, –3), (4, –7) and (6, –7) respectively, find by construction the centre and angle of rotation that maps ∆ ABC onto ∆ A′B′C′.

9. Under a rotation, the images of points P(–1,1) and Q (2,4) are P′(3, – 4) and Q′ (0, –1). Plot these points on a squared paper and find:

(a) The co–ordinates of the centre of rotation.

(b) The co–ordinates  of R given that   R′ is (–2, 4).

10. A (–1, –1), B(3, –1), C(3, 3), D(–1, 3),  E(–3, –3), F(1, –3), G (1, 1) and H(–3, 1) are the vertices of two squares ABCD and EFGH. Draw the squares on squared paper.

(a) Find the centre and the angle of rotation that maps:

(i) ABCD onto FGHE,

(ii) EFGH onto DABC.

(b) What is the equation of the mirror line of the reflection that maps one square onto the other?

11. State which of the following statements are true and which are false. When a figure or object has a translation applied to it,

(a) all points move in the same direction.

(b) not all points of the figure move in the same direction.

(c) all lengths in the object remain unchanged.

(d) usually, at least one point on the figure remains unchanged.

(e) a translation can be described by many directed line segments, provided each has the same length and same direction.

12. (a) Triangles ABC, PQR and STU are  congruent. Co–ordinates of the vertices of the triangles are given as A(1, 1), B(4, 3), C(1, 2); P(–1, 1), Q(–4, 3), R(–1, 2); S(– 4, 4), T(1, 6) and U(–4, 5).  On the same axis plot the points and draw the triangles. Describe the congruence between the following triangles:

(i)  ABC and PQR.

(ii) ABC and STU.

(b) Describe the transformation that maps:

(i) ABC onto PQR

(ii) ABC onto STU.

13. Find the image of ∆ABC, where A is  (–3, –2),  B is (–1, 1)  and  C is (2, –1), with operation vector 4 2 .

14. The image of A (6, 4) under a translation is A′ (3, 4). Find the translation vector.

15. ∆ABC with A (0, 1),  B (2, 0) and C (3, 4) is given a translation equivalent to 2 4 followed by  –3 2 .      Find the coordinates of the image of  ∆ABC.

16. Quadrilateral P′Q′R′S′ is the image of PQRS under a certain translation.   P is  (2, 0),  Q is (0, 3),  P′ is (5, 4),  R′ is  (6, 9) and S′ is (9, 4).  Find:

(a)  the translation vector.

(b)  the coordinates of R, S and Q′.