KT Classroom: 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

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: A( x; y ) → A ' ( x ' ; y ' ) 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 90 o •

When a point is rotated clockwise about the origin through 90 o its coordinates change

•

When a point is rotated anti‐clockwise about the origin through 90 o its coordinates change

according to the rule ( x; y ) → ( y;− x )

according to the rule (x; y ) → (− y; x )

Rotation about the origin through 180 o This is the same as two successive rotations about the origin through 90 o

Enlargement through the origin by a factor k Each point is multiplied by k A( x; y ) → A' ( kx; ky )

area after transformation 2 = (k ) area before transformation

and

3

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ROTATION OF A POINT ABOUT THE ORIGIN THROUGH AN ANGLE OF θ o Sketch x and y ‐ axes in the space opposite.

Mark a point A with coordinates ( x; y ) 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 β o 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:

cos β =

x y and sin β = r r

so that

x = rcos β and y = rsin β

Now A is rotated through an angle θ o to a point A′(x′; y ′) or the line OA to OA′ . The direction of the rotation could be clockwise or anti‐clockwise. It does not matter which direction nor does the size of angle θ o but make it anti‐clockwise and into the second quadrant.

In the space opposite, sketch both OA and OA′ on the same set of axes showing the angles β o and θ o .

From A ′ drop a perpendicular to the x ‐axis.

So you now will have

x′ = rcos(β + θ ) and y ′ = rsin(β + θ )

Formula for rotation of a point about the origin Apply compound angle formulae to these last equations aboveto show that A′( x ′; y ′) ≡ A' ( x cos θ − y sin θ ; y cos θ + x sin θ ) which gives the coordinates of A after rotation about the origin through an angle of θ o .

Transformation Geometry

PAPER 2 QUESTION 3

DoE/ADDITIONAL EXEMPLAR 2008

5

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PAPER 2 QUESTION 3

Number Hints and answers 3.1.1 Read the revision notes on

transformation on page 98

3.1.2 Read the revision notes on

transformation on page 98

3.1.3 Read the revision notes on

transformation on page 98

3.2.1 Read the revision notes on transformation on page 98

3.2.2 Read the revision notes on & 3.2.6 transformation on page 98

Use the opposite grid to answer Questions 3.2.2 and 3.2.6.

3.2.3

Read the revision notes on transformation on page 98

3.2.4

Read the revision notes on transformation on page 98

3.2.5

Read the revision notes on transformation on page 98

DoE/ADDITIONAL EXEMPLAR 2008

Write down the answers in the boxes below

Answer:

(x; y ) →

DIAGRAM SHEET 1

A’C’=

Area of Δ A’B’C’ =

=

=

A’’( ; ) =

Transformation Geometry

PAPER 2 QUESTION 3

DoE/NOVEMBER 2008

7

Preparation for the Mathematics examination brought to you by Kagiso Trust

PAPER 2 QUESTION 3

Number Hints and answers

3.1.1 Read the revision notes on transformation on page 98

3.1.2 Read the revision notes on transformation on page 98

3.2.1 Read the revision notes on transformation on page 98

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

DIAGRAM SHEET 1

3.2.3

3.2.4

3.2.5

Read the revision notes on transformation on page 2

Read the revision notes on transformation on page 2

Read the revision notes on transformation on page 2

DoE/NOVEMBER 2008

Write down the answers in the boxes below

Transformation Geometry

MORE QUESTIONS FROM PAST EXAMINATION PAPERS Exemplar 2008

9

Preparation for the Mathematics examination brought to you by Kagiso Trust

Transformation Geometry

Preparatory Examination 2008 QUESTION 3

11

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November 2008

Feb – March 2009

Transformation Geometry

13

Preparation for the Mathematics examination brought to you by Kagiso Trust

November 2009 (Unused paper)

Transformation Geometry

November 2009 (1)

15

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Feb – March 2010

Transformation Geometry

17

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ANSWERS

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

Exemplar 2008

( ) 3.1.2 P(− 3;2)

3.1.1 P 2;− 3

3.1.2 A’B’ = 2 29 3.1.3 Shape and size remain the same under a translation so the length AB =length A’B’.

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

3.2.1 Sketch:

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

3 1 − y⋅ 3.3.1 x' = x ⋅ 2 2 3 1 + x⋅ 3.3.2 y ' = y ⋅ 2 2

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

⎛

3.2 A’ ⎜⎜ 3 +

⎝

1 3⎞ ⎟ or A’(2,23; ‐0,13) ;−1 + 2 2 ⎟⎠

Transformation Geometry

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

3.2 R(− 1;3) → R' ' (2;7)

Feb/March 2009

( 3.1.2 P ' (−

) 3 ;2 )

3.1.1 P' − 3;−2

3.3 ( x; y ) → ( y; x) or reflection about the line y = x .

3.2.1 Q' (2;2) 3.2.2 Sketch:

3.4 θ = 90 o a rotation in an anti‐clockwise direction; or o θ = 270 a rotation in a clockwise direction.

4.1 θ = 36,03 o

⎛3 3 ⎞ 3 + 2;− + 2 3 ⎟⎟ 2 ⎝ 2 ⎠

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

3.2.5 ( x; y ) → (2 y;−2x) 3.2.6

Area PQRS 1 = P' ' Q' ' R' ' S' ' 4

⎛

2 2 ⎞ x− y⎟ 2 ⎟⎠ ⎝ 2 ⎛ 2 2 ⎞ y ' = ⎜⎜ − y+ x ⎟⎟ 2 2 ⎝ ⎠ 4.2 M − 3 2 ;− 2 4.1 x' = ⎜⎜ −

(

)

19

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November 2009(1) 6.1.1 20 sq units 6.1.2 ( x; y ) → (2x;2 y ) 6.1.3 Sketch:

Feb/March 2010 7.1.1 P' (5;−2)

7.1.2 P' (5;2)

7.2.1 Reflection across the line y = x followed by contraction by scale

factor of

1 ; or vice versa.

2

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

⎛ −3− 2 3 3 3 − 2⎞ ⎟ ; ⎟ 2 2 ⎝ ⎠

8.1 P' ⎜⎜

(

)

8.2 Q − 2;2 3

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 (x; y) → (x − 2;− y − 3)

7.1 T ' ( x cosθ − y sinθ ; y cosθ + x sinθ )

(

)

7.2 A' p cos135o − q sin 135o ; q cos135o + p sin 135o if clockwise rotation.

(

)

7.3 A 2 ;2