3.1B Determining the Image of an Object Under Combination of Enlargement and

Isometric Transformation

In the diagram, rectangle A is an object.Given:

E is an enlargement at (0,2) with a scale factor of 2R is a rotation through 90 clockwise at (-1, -1)P is a reflection in the line y = 5, andT is a translation .

Determine the image of rectangle A under the following combined transformations of:a) E2 b) ER c) PE d) ET

Example:

Solution:

a) Transformation E2 (E is an enlargement at (0,2) with a scale factor of 2 followed by another E).

The image of rectangleA under the combinedtransformation of E2 isrectangle I.

b) Transformation ER, (R is a rotation through 90 clockwise at (-1, -1) followed by an enlargement E at (0,2) with a scale factor of 2)

The image of rectangle Aunder the combined transformation of ER is rectangle III.

c) Transformation PE ( E is an enlargement at (0,2) with a scale factor of 2 followed by a reflection P in the line y = 5).

The image of rectangle Aunder the combined transformation of PE is arectangle IV.

d) Transformation ET (T is a transformation followed by an enlargement E at (0,2) with a scale factor of 2).

The image of rectangle A under the combined transformation of ET is rectangle II.

3.1 E Determining the Equivalent of a Combined Transformation

When a transformation A is combined with a transformation B, we can determine whether the combined transformation AB is equivalent to the combined transformation BA by comparing the images obtained from AB and BA. There are cases where AB is equivalent to BA and there are also cases where AB is not equivalent to BA.

Example 1:

The transformation T and E are defined as follows:T = Translation E = Enlargement with a scale factor of 3 and the origin as

the centre(a) Find the coordinates of the image of the point A (-3, 2)

under(i) the combined transformation ET(ii) the combined transformation TE

(b) Is ET equivalent to TE?

Solution:

(a) (i)

A(-3, 2) A’(1, 1) A’’(3, 3)

Hence, the coordinates of the image of the point A(-3, 2) under the combined transformation ET is (3, 3).

T E

(ii)

A(-3, 2) A’(-9, 6) A’’(-5, 5)

Hence, the coordinates of the image of the point A(-3, 2) under the combined transformation TE is (-5, 5).

(b) ET is not equivalent to TE.

E T

Example 2:

The transformation P and E are defined as follows:P = Reflection in the x-axisE = Enlargement with centre O

(0, 0) and a scale factor of 2(a) Draw the image of the figure

ABCD in the diagram under(i) the combined

transformation EP(ii) the combined

transformation PE(b) Is EP equivalent to PE?

Solution:

(a) (i)

Figure ABCD figure A’B’C’D’ figure A’’B’’C’’D’’P E

(ii)

Figure ABCD figure A’B’C’D’ figure A’’B’’C’’D’’

(b) EP is equivalent to PE.E P

3.1 H Solving Problems Involving Transformations

RECALL

1) Scale factor, k =

2) k2 =

Example:

In the diagram, the triangle ACD is the image of triangle ABC under an enlargement followed by another transformation P.Given, AB = 4 cm, AC = 8 cm and CD = 12 cm.a) State the scale factor for the enlargement.b) Calculate the length of BC.c) Describe precisely transformation P.

Solution :

a) AB will be transformed to AC.Scale factor =

= = 2

The scale factor for the enlargement is 2.b) The scale factor is 2 and BC is transformed to CD.

Therefore,2BC = CD2BC = 12 cm BC = 6 cm

c) AB’C is the image of ABC (under an enlargement).AB’’C’’ is the image of AB’C’ (under a rotation).

Transformation P is a rotation through 45 clockwise at A.