﻿ Matrices and Transformations II

# KCSE ONLINE

## One stop Education solutions

### Revision

Other

Candidate benefit from our revision notes which are comprehensive and show how to tackle examination questions effectively

### Results

Moreover

As a supplementary to coursework content our e-library for digitized multimedia CDs while enhance and ensure that you never missed that important concept during the normal class lessons. It is a Do it Yourself Project

### KNEC

For Best results INSTALL Adobe Flash Player Version 16 to play the interactive content in your computer. Test the link below to find out if you have Adobe Flash in your computer

## Matrices and Transformations II

REFLECTION

Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

REFLECTION

Under reflection, the object and its image have the same size and shape. The object and its image are equidistant from the mirror line. In addition, a straight line that joins a point on the object and its image is perpendicular to the mirror line. This is shown in the following diagram.

M is the mirror line
QQ' is perpendicular to line M
QN = NQ'
Triangle PQR and P'Q'R' have the same shape and size.

ROTATION

Under rotation the object and the image have the same size and shape. The process of rotation involves identification of the centre, angle and direction of rotation.In the above diagram triangle P'Q'R' is the image of triangle PQR under rotation of 95o about O in a clockwise direction. PQR is said to have been rotated through-95o about O.
If triangle P'Q'R' is rotated in an anticlockwise direction to get triangle PQR, then it is said to have been rotated through 95
Note that the size and shape of triangles PQR and P'Q'R' are the same.

ENLARGEMENT

Under enlargement the size of an object, increases or decreases in a given ratio. When an object is enlarged:

• Sides of the object are parallel to their corresponding sides on the image

• Lengths of the sides of an object and lengths of their images are in the same ratio.

• Angles do not change

In the illustration below, line AB is parallel to A'B'.

Angle ABC = Angle A'B'C'
Angle BCD = Angle B'C'D'

LINEAR SCALE FACTOR OF ENLARGEMENT

When the linear scale factor is greater than positive one (+ve 1), the image is larger and on the same side of the centre of enlargement as the object. When the linear scale factor is less than +ve 1, but greater than 0 the image is smaller and on the same side of the centre of enlargement as the object.
When the scale factor of enlargement is negative, the centre of enlargement lies between the object and the image. In this case, the image is inverted as shown below

Triangle ABC is enlarged using linear scale factor of -2 centre of enlargement O.

MATRICES AND TRANSFORMATIONS II
Introduction
Life is full of changes both the living and the non living change. Transformation involves changing something in some way. In Mathematics the knowledge of transformation helps us to understand how points and objects change in terms of positions, shape, size and direction.

Transformations

Transformation on a Cartesian plane
The following graphs show various types of transformations. Describe fully the transformation that maps:
Solutions

1.Reflection in the line y = 0
2.Enlargement scale factor positive 2 centre (0,0)
3.Rotation positive 90
4.Translation using translation vector (-7)

Determining the matrix of transformation.
We can obtain a matrix of transformation using the coordinates of the object and its image.
Example 1
A square with vertices O(0,0), A (1,0), B(1,1) and C(0,1) is mapped onto the image O'(0,0), A' (-1,0), B'(-1,1) and C'(0,1). Determine the transformation matrix that maps OABC onto O'A'B'C'.

Solution

Example 2

Plot the coordinates of the image on the graph below by clicking on the vertices of the image.

The transformation is a reflection in the line x = 0

Example 3
a) Use the graph below to determine the matrix that maps:
i) PQR onto P'Q'R'
ii) P'Q'R'onto P''Q''R''
iii) P''Q''R'' onto P'''Q'''R'''
b) Determine a single matrix of transformation that maps PQR onto P''Q''R''

b) A single matrix of transformation that maps PQR and P''Q''R''can be obtained in two ways, namely:
i) multiplying the matrix that maps PQR onto P'Q'R' by the matrix that maps P'Q'R' onto P''Q''R''.
Let the matrix that maps PQR onto P'Q'R' be A and the one that maps P'Q'R' onto P''Q''R'' be B.

The single matrix of transformation is

Matrix and equate to the final image matrix and work out unknowns a, b, c and d. That is

So that a = 2, b = 0, c = 0 and d = -2

Relationship between area scale factor and determinant of a matrix
Under the transformation of enlargement, the ratio of the sides of the image and the corresponding sides of the object is constant. This is referred to as the linear scale factor. The square of the linear scale factor gives the area scale factor. The determinant of the matrix of transformation is equal to the area scale factor.

NB, the area scale factor can also be obtained by dividing the area of the image by the area of the object.

Example
Use the following graph to determine:

i)The area of triangle KLM
ii)The area of triangle K'L'M'
iii)Area scale factor.
iv)The matrix P that maps triangle KLM onto K'L'M'.

V)The determinant of matrix P
Vi) State the relationship between area and scale factor and the determinant of P.

STRETCH
Stretch is a transformation in which all the points of an object move at right angles to an invariant (i.e. fixed) line. The distance moved by a given point from the fixed line is proportional to its original distance from the line
Example
Use the following graph to;

I.Identify the fixed (invariant) line
II.Determine the matrix p, that maps ABCD onto A'B'C'D'
III.Describe the transformation.

Solution
i. The invariant line is y-axis (x=0)

Note: the matrix of the vertices that moves is pre-multiplied by the transformation matrix and the product equated to the matrix of the corresponding image.

The points on the invariant line do not move and therefore should not be used when working out the transformation matrix.

SHEAR
A shear is a transformation in which all points of an object move parallel to a fixed or invariant line. The distance moved by a given point is proportional to the distance from the invariant line.

Example
Use the following graph to

i.Identify the fixed (invariant) line
ii.Determine the matrix Q that maps ABCD onto A'B'C'D'
iii.Describe the transformation that maps ABCD onto A'B'C'D'

SOLUTION
I.The invariant line is x axis (y=0)

Note: the matrix of the vertices that moves is pre-multiplied by the transformation matrix and the product equated to the matrix of the corresponding image.

.

Matrices and Transformations II

## e-Content

Buy e-Content Digital CD covers all the topics for a particular class per year. One CDs costs 1200/-

click to play video

Purchase Online and have the CD sent to your nearest Parcel Service. Pay the amount to Patrick 0721806317 by M-PESA then provide your address for delivery of the Parcel.. Ask for clarification if in doubt,

Matrices and Transformations II

###### We have an enourmous data quiz bank of past papers ranging from 1995 - 2017

KCSE ONLINE WEBSITE provide KCSE, KCPE and MOCK Past Papers which play a great role in students� performance in the KCSE examination. KCSE mock past papers serves as a good motivation as well as revision material for the major exam the Kenya certificate of secondary education (KCSE). Choosing the KCSE mock examination revision material saves you a lot of time spent during revision for KCSE . Choosing the KCSE mock examination revision material saves you a lot of time spent during revision for KCSE. It is also cost effective

#### MOCK Past Papers

As a student, you will have access to the most important resources that can help you understand what is required for you to sit and pass your KCSE examination and proceed to secondary school or gain entry to University admission respectively.

#### KCSE ONLINE

Similar

More

Similar

KCSE ONLINE WEBSITE provide KCSE, KCPE and MOCK Past Papers which play a great role in students� performance in the KCSE examination.

Choosing the KCSE mock examination revision material saves you a lot of time spent during revision for KCSE. It is also cost effective

Ask for clarification if in doubt, vitae dignissim est posuere id.

Matrices and Transformations II

sit amet congue Mock Past Papers, give you an actual exam situation in readiness for your forthcoming national examination from the Kenya National Examination Council KNEC

Choosing the KCSE mock examination revision material saves you a lot of time spent during revision for KCSE. It is also cost effective sapien.

Choosing the KCSE mock examination revision material saves you a lot of time spent during revision for KCSE. It is also cost effective sapien.

Matrices and Transformations II

As a supplementary to coursework content our e-library for digitized multimedia CDs while enhance and ensure that you never missed that important concept during the normal class lessons. It is a Do it Yourself Project..

Candidates who would want their papers remarked should request for the same within a month after release of the results. Those who will miss out on their results are advised to check with their respective school heads and not with the examination council

For Best results INSTALL Adobe Flash Player Version 16 to play the interactive content in your computer. Test the link below to find out if you have Adobe Flash in your computer.

### New Register

Similar

Buy e-Content Digital CD covering all the topics for a particular class per year. One CDs costs 1200/- ( Per Subject per Class )

### New Membership

Also

We have an enourmous data quiz bank of past papers ranging from 1995 - 2017

### Register

Gold

Register as gold member and get access to KCSE and KCPE resources for one year. subscription is 1000/- renewable yearly.