transformation geometry

MATHEMATICS

Learner’s Study and

Revision Guide for

Grade 12

TRANSFORMATIONS

Revision Notes, Exercises and Solution Hints by

Roseinnes Phahle

Examination Questions by the Department of Basic Education

Preparation for the Mathematics examination brought to you by Kagiso Trust

Contents

Unit 15

Revision notes 3

Examination questions with solution hints and answers 5

More questions from past examination papers 9

Answers 18

How to use this revision and study guide

1. Study the revision notes given at the beginning. The notes are interactive in that in some parts you are required to make a response based on your prior learning of the topic in class or from a textbook.

2. “Warm‐up” exercises follow the notes. Some exercises carry solution HINTS in the answer section. Do not read the answer or hints until you have tried to work out a question and are having difficulty.

3. The notes and exercises are followed by questions from past examination papers.

4. The examination questions are followed by blank spaces or boxes inside a table. Do the working out of the question inside these spaces or boxes.

5. Alongside the blank boxes are HINTS in case you have difficulty solving a part of the question. Do not read the hints until you have tried to work out the question and are having difficulty.

6. What follows next are more questions taken from past examination papers.

7. Answers to the extra past examination questions appear at the end. Some answers carry HINTS and notes to enrich your knowledge.

8. Finally, don’t be a loner. Work through this guide in a team with your classmates.

Transformation Geometry

3

REVISION UNIT 15: TRANSFORMATION GEOMETRY

The transformation rules below do not appear on the information sheet given to you in the examination. The best way to get to know them is by practicing them on a point, say, (5; -3) or any point you like.

There are four types of transformations that must understand. These are

• Translation (which means a shift) • Reflection (as in a mirror) • Rotations (turning around the origin in a clockwise or anti-clockwise direction) • Enlargement (through the origin by a constant factor k )

Notation: );();( ''' yxAyxA → where 'A is the image of A under the transformation.

Translations

p units horizontally: (x; y)→ (x+p; y) q units vertically: (x; y)→ (x; y+q) p units horizontally and q units vertically: (x; y)→ (x+p; y+q) Reflection

About the x‐axis: (x; y) → (x; ‐y) About the y‐axis: (x; y) → (‐x; y) About the line y = x: (x; y) → (y; x) Rotation about the origin through o90

• When a point is rotated clockwise about the origin through o90 its coordinates change

according to the rule ( ) ( )xyyx −→ ;;

• When a point is rotated anti‐clockwise about the origin through o90 its coordinates change

according to the rule ( ) ( )xyyx ;; −→

Rotation about the origin through o180

This is the same as two successive rotations about the origin through o90

Enlargement through the origin by a factor k

Each point is multiplied by k

and );(');( kykxAyxA → ( )2

ationtransformbeforeareasformationafter tran area k=


ROTATION OF A POINT ABOUT THE ORIGIN THROUGH AN ANGLE OF oθ

Sketch x and y ‐ axes in the space opposite.

Mark a point A with coordinates ( )yx; in the first quadrant (but the point could be in any quadrant). Let the distance of A from the origin be r units. That is OA=r . Let OA make an angle of oβ with the positive direction of the x ‐axis. From A drop a perpendicular line to the x ‐axis. Looking at your diagram you have that:

cosrx

=β and sinry

=β

so that βcosrx = and βsinry =

Now A is rotated through an angle oθ to a point ( )yx ′′′ ;A or the line OA to AO ′ . The direction of the

rotation could be clockwise or anti‐clockwise. It does not matter which direction nor does the size of angle

oθ but make it anti‐clockwise and into the second quadrant. In the space opposite, sketch both OA and AO ′on the same set of axes showing the angles oβ and oθ . From A′ drop a perpendicular to the x ‐axis. So you now will have ( )θβ +=′ cosrx and ( )θβ +=′ sinry

Formula for rotation of a point about the origin

Apply compound angle formulae to these last equations aboveto show that

)sincos;sincos(');( θθθθ xyyxAyxA +−≡′′′

which gives the coordinates of A after rotation about the origin through an angle of oθ .


5

PAPER 2 QUESTION 3 DoE/ADDITIONAL EXEMPLAR 2008


PAPER 2 QUESTION 3 DoE/ADDITIONAL EXEMPLAR 2008

Number Hints and answers Write down the answers in the boxes below 3.1.1

Read the revision notes on transformation on page 98

3.1.2


3.1.3


3.2.1 Read the revision notes on transformation on page 98

Answer: ( )→yx;

3.2.2 &

3.2.6

Read the revision notes on transformation on page 98 Use the opposite grid to answer Questions 3.2.2 and 3.2.6.

DIAGRAM SHEET 1

3.2.3


A’C’=

3.2.4


Area of Δ A’B’C’ = = =

3.2.5


A’’( ; ) =


7

PAPER 2 QUESTION 3 DoE/NOVEMBER 2008


PAPER 2 QUESTION 3 DoE/NOVEMBER 2008

Number Hints and answers Write down the answers in the boxes below 3.1.1 Read the revision notes on

transformation on page 98



3.2.2 Sketch the polygon A’B’C’D’E’ here: DIAGRAM SHEET 1





9

MORE QUESTIONS FROM PAST EXAMINATION PAPERS

Exemplar 2008


11

Preparatory Examination 2008

QUESTION 3


November 2008

Feb – March 2009


13


November 2009 (Unused paper)


15

November 2009 (1)


Feb – March 2010


17


ANSWERS

Exemplar 2008

3.1.1 ( )3;2 −P

3.1.2 ( )2;3−P 3.2.1 Sketch:

3.2.2 ( ) ( )yxyx 2;2; −−→ 3.2.3Area of ABCD : Area of PQRS = 1 : 4

3.3.1 21

23' ⋅−⋅= yxx

3.3.2 21

23' ⋅+⋅= xyy

3.4 K’(1,96; 4,60) and L’(‐0,40; 6,70)

Preparatory Examination 2008 3.1.1 C’(‐2; ‐4)

3.1.2 A’B’ = 292 3.1.3 Shape and size remain the same under a translation so the length AB =length A’B’.

3.1.4(a) ( ) ( )xyyx 5,0;5,0; → 3.1.4(b) Sketch:

3.2 A’ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

231;

213 or A’(2,23; ‐0,13)


19

Feb/March 2009

3.1.1 ( )2;3' −−P

3.1.2 ( )2;3' −P

3.2.1 ( )2;2'Q 3.2.2 Sketch:

3.2.3 ( )6;4''P 3.2.4 Shape remains the same but the size changes. Thus not rigid.

3.2.5 ( ) ( )xyyx 2;2; −→

3.2.6 41

'S''R''Q''P' Area

=PQRS

4.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−−= yxx

22

22'

⎟⎟⎠

⎞⎜⎜⎝

⎛+−= xyy

22

22'

4.2 ( )2;23 −−M

November 2009 (Unused paper) 3.1 Enlargement by scale factor 2.

3.2 ( ) ( )7;2''3;1 RR →−

3.3 ( ) ( )xyyx ;; → or reflection about the line xy = .

3.4 o90=θ a rotation in an anti‐clockwise direction; or

o270=θ a rotation in a clockwise direction.

4.1 o03,36=θ

4.2 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−+ 32

23;2

233''P


November 2009(1) 6.1.1 20 sq units

6.1.2 ( ) ( )yxyx 2;2; → 6.1.3 Sketch:

6.1.4 Transformation is not rigid because only the shape remains the same but the size changes. 6.2 Translates 2 units to the left and 3 units down or

( ) ( )3;2; −−−→ yxyx

7.1 ( )θθθθ sincos;sincos' xyyxT +−

7.2 ( )oooo 135sin135cos;135sin135cos' pqqpA +− if clockwise rotation.

7.3 ( )2;2A

Feb/March 2010

7.1.1 ( )2;5' −P

7.1.2 ( )2;5'P 7.2.1 Reflection across the line xy = followed by contraction by scale

factor of 21; or vice versa.

7.2.2 ( )4;8'H or ( )16;8'H 7.2.3 Area KUHLE:Area K’’U’’H’’L’’E’’=4:1

8.1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−2

233;2

323'P

8.2 ( )32;2−Q

transformation geometry

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