Gr 11 Kickass Maths final · 2020-07-24 · 1 Kickass Maths - Introduction and examination guidelines You must know ALL theorems and geometry facts. You must be able to apply ALL

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ck-kk-Geometry

Solutions of Geometry problems: M Smit & M van NiekerkEditing: C Oosthuizen

Copy rightThe content of this book is the intellectual property of Morné Smit.

Unauthorized duplication of any kind can result in criminal prosecution.

Compiled by M. Smit

Published by:

Tel: 014 592 6083Cell: 079 092 0519(no sms Vodacom)/063 133 6292(no sms MTN)Email: [email protected]/ www.amaniyah.co.zaISBN 978-1-928528-08-1eISBN 978-1-928528-16-6

Help us to deliver a better product with each print by sending suggestions to:

[email protected].

First edition 2019

Thank you very much for purchasing this Geometry workbook. It will definitely contribute to the success of

your academical future. I trust that you will gain a lot from it.

This workbook was specifically compiled so that you can enjoy Geometry and to make it easier to

understand. After many years’ of experience I know what difficulties learners encounter and I believe that

my way of teaching will be to the benefit of all. If you diligently work through this workbook, you will achieve

success in tests and exams.

Mathematics is not a subject for spectators. If you spend all your time watching the teacher and fellow

learners doing it, you will never achieve good results in Mathematics. You must be actively involved and

solve the problems by YOURSELF. I always tell my learners: “You may have the best Mathematics teacher

in the whole wide world, but if you do not practice it on YOUR OWN, you will never be able to do it by

YOURSELF”.

The practical understanding of Mathematics is partly theory that you need to know by heart. If you do not

understand something it is simply because there are certain sections of theory (laws, statements etc.) that

you do not know well enough. Be sure that you know your theory at all times.

Good luck with your Mathematics at school. Remember that Mathematics will unlock many doors for you

and it will ensure you of a bright future.

Greetings

Morné Smit

This books is dedicated to Henna Cornelius - a very special grandmother who always encouraged me to

live my dreams!

PREFACE

More about the author...

Mr. Morné Smit has more than 15 years of experience in

the successful teaching of Mathematics. He has been an

examiner, a moderator, chief marker & deputy chief marker

for many years for the grade 11 & 12 external examinations in

the North West province. In 2015 Mr Smit won the National

Teaching Awards and was nominated as the best Mathematics

teacher in South Africa.

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1

2

3

4

5

6

7

8

INDEX

Angles inside a circle: Angles at Circumference, Angles at the centre and Angles in a semi-circle

Introduction and examination guidelines

Revision of previous geometric concepts

The theorem of Pythagoras

The lines inside a circle

Cyclic quadrilaterals

Solutions

Tangents to circles

1page 1

page 7

page 17

page 21

page 34

page 53

page 76

page 111

2

3

4

5

6

7

8

Kickass Maths - Introduction and examination guidelines1

You must know ALL theorems and geometry facts.

You must be able to apply ALL theorems and geometry facts in diagrams.

In every problem you must give your OWN statements and then the correct reasons

to support it.

You must prove four theorems flawlessly in tests and examinations. Make sure you always learn your statements first! These are give-away marks!

In grade 11-circle geometry the following will be expected of you:

Euclidean Geometry means that you never

measure the angles or sides, but that you must

use GEOMETRY-THEORY (theorems and facts)

to answer the questions.

Therefore you must never contemplate

measuring the sides with a ruler or the angles

with a protractor in order to find the answers to a

geometry problem.

INTRODUCTION

Fact 1

Fact 2

Fact 3

Fact 4

Do not make geometry unnecessarily difficult for

yourself. If you know all the geometry-theorems

and facts, then geometry is a much easier

section than Algebra.

When geometry questions are answered, there

must be a STATEMENT (that you yourself have

calculated or derived) and a REASON (which

supports your calculation or derivation) given.

The following marking guidelines are very important, because your teacher will be applying them strictly every time he/she marks your Geometry test or examination papers:

Fact 1 Your statement MUST BE CORRECT otherwise you cannot get a mark for the reason. You get no marks for a correct reason if the statement is wrong. Therefore give a lot of attention to the correct statements.

You may never allocate a specific size to an angle. For example, you may not state that an angle is 30° etc. You may however, let an angle be equal to x (or any other variable).

Fact 2

Fact 3 You will always get a theorem to prove. Always make sure that your construction line is correct, otherwise you will get NO marks for the theorem. It is known as a “break down ” in Mathematics.

(The prescribed theorems of which the formal proofs must be known for examination purposes, are given and proven at the end of each module).

Geometry facts/theorems that were taught in grade 11, usually get two marks (�statement and �reason) if it is used in a problem. All previous facts/theorems (gr.8 – 10) get only one mark when it is used in a geometry problem.

Fact 4

2Kickass Maths - Introduction and examination guidelines

Terminology Symbol / Abbreviation

AngleAngles sAngle A ÂAngle A and Angle B are 180° Â+ B̂ =180°Triangle �Circle Circle with centre A AVertically opposite angles vert opp. sOpposite angles opp. sPerpendicular Parallel IICyclical quadrilateral cyclic quad.Parallelogram Parm or IIm

Similar triangles ��Midpoint Midpt.Corresponding angles Corresp. sAlternate interior angles Alt.int. s

T

ACCEPTABLE ABBREVIATIONS AND SYMBOLS IN GEOMETRY#1

Study the use of the following symbols very well:

Kickass Maths - Introduction and examination guidelines3

When you give reasons, there are certain acceptable abbreviations or symbols you may use. Below is a couple of these acceptable abbreviations and symbols:

Theorem Acceptable reason 1. If two straight lines intersect one another, the vertically

opposite angles are equal. vert opp s

2. The sum of angles on a straight line is 180°. 's on straight line

3. The sum of the interior angles of a triangle is 180°. inside ��

4. The exterior angle of a triangle is equal to the sum of the interior opposite angles.

outside ��

5. If AB II CD then the corresponding angles are equal. corresp. s [AB II CD]

6. If AB II CD then the CO-interior angles are supplementary. co-int s [AB II CD]

7. f AB II CD then the alternate angles are equal. alt s [AB II CD]

Grade 8 - Geometry

Grade 10 - Geometry

Theorem Acceptable reason 8. The line segment joining the midpoints of two sides of a

triangle is parallel to the third side and equal to half the length of the third side.

midpt.- theorem

9. Two triangles on the same base and between the same parallel lines, have equal surfaces.

same base; same height/ equal .

bases equal height. II lines

10 The sum of the interior angles is 180°. interior ��

11. The opposite sides of a parallelogram are equal in length. opp. sides of parm

12. The opposite sides of a parallelogram are parallel. opp sides of parm

13. The opposite angles of a parallelogram are equal. opp 's of parm

14. The diagonals of a parallelogram bisect each other. diag of parm

15. The diagonals of a parallelogram bisects area. diag bisect area of parm

Theorem Acceptable reason 16. The line drawn from the centre of a circle perpendicular to a

chord, bisects the chord. line from centre to chord

17. The line drawn from the centre of a circle to the midpoint of a chord, is perpendicular to the chord.

line centre. ; centre chord

18. The angle at the center of a circle is twice the circumference

angle. centre = 2× at

circumference

19. The angle in a semi-circle is 90°. in 1/2

20. Perimeter angles intersected by the same arc or chord are equal. (the BOW TIE).

's int the same seg

Grade 11 - Geometry

4Kickass Maths - Introduction and examination guidelines

21. The sum of the opposite angles of a cyclic quadrilateral, equals 180°.

opp. s of cyclic quad.

22. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

ext of cyclic quad.

23. The tangent to a circle is perpendicular to the radius. tan

T

radius

24. The angle between the tangent and a circle and a chord is equal to the angle in the alternate segment.

tangent - chord theorem / between tangent and chord.

25. Two tangents drawn to a circle from the same point outside the circle, are equal in length.

Tangent from same pt.

Grade 12 - Geometry

Theorem Acceptable reason 26. A line drawn parallel to one side of a triangle divides the

other two sides proportionally. (Assuming AB II CD )Prop. theorem [AB II CD]

or line II��

27. If two triangles are equiangular, then the corresponding sides are in proportion.

��

Write the following statements in an abridged version so that it is acceptable in all tests and exams. (1 is done as an example for you)

Geometric statements Acceptable shortened version1. The opposite angles of a parallelogram The opp. 's of a parm2. The central angle of a circle 3. Triangle ABC and triangle PQR are similar

triangles 4. The interior angles of a cyclic quadrilateral 5. The opposite angles of a cyclic

quadrilateral 6. AB is perpendicular to CD 7. PQ is parallel to TR 8. Line KM is twice the length of LP 9. Line AB is half the length of line CD 10. Alternate interior angles 11. Corresponding angles 12. Angle A is the same size as angle B 13. Angle P is three times the size of angle M 14. Angle X and angle Y together equal 120° 15. The circle with centre O

Exercise 1 1 2 365

4 987

+

-x

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c

.0 =

A Kick ass Maths - Revision of basic geometry5

REVISION OF BASIC GEOMETRY #2

How do we name angles?

DA

BC

�1 2

�

AÊD = �Lines AE and ED form the angle �. The angle is formed at E and therefore there is a cape on the E. (Slide your finger from A to E to D… The letter in the middle (E) is where the angle is formed.)

CÊB = �

Lines CE and EB form the angle �. The angle is formed at E and therefore there is a cape on the E. (Glide your finger from C to E to B… The letter in the middle (E) is where the angle is formed.)

If the angles are numbered, then there is a shorter way of describing the angles:

Ê1 =

Ê2 =

AÊD + CÊB + Ê1 + Ê2 = 360°(revolution) / <'s around a point.

6Kick ass Maths - Revision of previous geometric concepts

Angles formed by two lines

Types of angles Description of angles

Correct geometric reason

DA

BC

E�

�

If two straight lines intersect each other, then the verti-cally opposite angles are equal.

A D = C Bvert opp. e

D

A B C

TOGETHER the ADJACENT angles formed on a straight line is 180° (supplementary angles).

+ = 180°

A D + D C= 180° e on a str. line

D

A B C

E

TOGETHER complementary angles equal 90°.

+ = 90°

D E + E C = 90°Complimentary e

A Kick ass Maths - Revision of previous geometric concepts7

Angels of a triangle

Types of angles Description of angles

Correct geometric reason

A

B C

+ + = 180°

Together the interior angles of a triangle equal 180°.

If you “cut” all the angles of a triangle, they fit together on a straight line.

+ + = 180°Int. s of �

A

B C

��

When two sides of a tri-angle are equal, then the angles opposite the equal sides, are equal.

In an isosceles triangle two sides are always equal.

= s opposite equal sides

A

B C

��

When all 3 sides of a triangle are equal, then all 3 angles are equal and each one is equal to 60°.

= = = 60°

= = = 60°Equilateral �


D

A

B C

Exterior angles of a figure The exterior angle of a figure is a concept with which most learners sometimes

struggle. The biggest reason for that is many learners only focus on the “exterior” part.

is an exterior triangle ��

DA

B C

Is NOT an exterior angle ��

D

A

B C

A D = B C + A C ( Ext �� )

= +

The exterior angle of any triangle or quadrilateral is the angle that is formed on the outside against one of the sides of the figure, by extending and straightening one of the sides of the figure (with a ruler).

��!��"��#�

+ + = 180° (Interior ��

��!��$��#�

+ = 180° ( 's ��

+ = + +

= +


Parallel lines

F��%��

>

> =

U��&��

> >

+ = 180°

N��

> >

=


How do we prove parallel lines?

A B

C D

E

G

H

F

)��*��&gles ?

��"��,-- ./,2)�23�� 's

�34�5�)4*�5�

��)

��,27 -. ��%9� :��

x (180 – x)

A

B

C

D

E G

H F

)��;��<>?@#��"��

�,&�2A -�,-� 's9�

34�B��43�5�x + (180° – x��5�180°

��)

��C��,&��9�� 's�<>?@��%%�

A B

C D

E

G

H

F

)��D��angles ?

��"��FA -2A ��2A -�,-� 's

34�5��43�5� ��)

��C��9��9� :��


1 pair of parallel opposite sides.

>

>1area = (sum of parallel sides) x

T

2

>

>

>> >>

2 pairs of opposite sides are parallel.

2 pairs of opposite sides are equally long.

2 pairs of opposite angles are the same sizes.

Diagonals bisect each other.

area = basis x

T

h

>>

>

>>>>

2 pairs of opposite sides are parallel.2 pairs of opposite angles are the same size.Diagonals bisect angles.Diagonals bisect each other at 90°.

area = basis x

T

h

>

>

>

>> >>

2 pairs of opposite sides are parallel.2 pairs of opposite sides are equal.All 4 angles are 90° each.Diagonals bisect each other and are equal.area =l x b

>

>

>

>>>>

2 pairs of opposite sides are parallel.

All 4 sides are the same length.

All 4 angles are 90° each.

Diagonals are the same length..

Diagonals bisect each other perpendicularly and then bisects into 45° angles.

area = l x l

2 pairs of adjacent sides are equal

1 pair of opposite angles are the same size.

Longest diagonal bisects angles. Longest diagonal bisects shorter diagonal perpendicularly.

>

1area = (diagonal1 x diagonal2) 2

Properties of quadrilaterals Trapesium Parallelogram

Rhombus Rectangle

Square Kite


A segment is the area formed between a chord and the circumference of the circle. A chord always divides a circle into a large and small segment.

A chord is a line segment from one point on the circumference to another point on the circumference.

An arc is a section of the circumference of the circle. A chord or radius of a circle usually indicates a specific arc.

A sector is the area which is formed between two radii and the circumference of the circle.

Centre

Diameter

Sector

Radius

Arc

Tangent Small Segment

Chord

Segment

Sector

Components of a Circle


#1

#2

4x

3x

2x

A

B C

Given �ABC with = 3x ; = 4x and = 2x Calculate the value of x and hence the size of each angle.

3x180° - 7x

>

>P

Q R S

T

In each of the following cases reasons must be given for your statements:

Statement Reason

=

=

=

Statement Reason

Given PQR with P = 3x and RPT = 180°-7x

QS ll PT

Express the following angles in terms of x:

a) PRS

b) PQR

c) PRQ

Exercise 21 2 365

4 987

+

-x

_..%

c

.0 =


In each of the following cases reasons must be given for your statements:

Statement Reason

=

=

Statements Reason

PTQ =

QPT =

S =

TPS =

SUP =

#3

#4

3x– 40°

120°– xA

B C

DGiven parallelogram ABCD with

B =3x-40° and D =120°-x.

Calculate the value of:

a) x

b) B

c) A

P S

U

TQ Rx

Given parallelogram PQRS with QR extended to T. Line PT is drawn so that PQ = PT. Q = x.Express the following angles in terms of x:

a) PTQ

b) QPT

c) S

d) TPS

e) SUP

A Kick ass Maths - Revision of basic geometry15

The Theorem of Pythagoras (grade 8)

Rectangular side2 + Rectangular side2 = Hypotenuse2

(AB)2 + (BC)2 = (AC)2

PYTHAGORAS#3A

B C

9

16

3 5

4

25

! B/DIt is VERY important to write down the theorem of Pythagoras correctly. If you formulate the theorem incorrectly, it is seen as a breakdown (B/D) and you would lose all the marks for that specific question.

16Kickass Maths - The theorem of Pythagoras

Solve x in the following quadratic equation: x2 + (x – 1)2 = (x + 1)2

Solution:x2 + x2 – 2x + 1 = x2 + 2x + 1x2 - 4x = 0x(x – 4) = 0x = 0 of x = 4

Determine the value of x in the right-angled triangle below.

(AB)2 + (BC)2 = (AC)2 … Pythagoras (x)2 + (x – 1)2 = (x + 1)2 … Every lateral length in its OWN bracket x2 + x2 – 2x + 1 = x2 + 2x + 1x2 - 4x = 0x(x – 4) = 0x = 0 of x = 4But x is a LATERAL LENGTH and CANNOT be negative or zero. Only x = 4

A

CB

x+1x

x-1

A Kickass Maths - The theorem of Pythagoras17

#1 #2A

B C M

T

K

2x

x + 3

63x - 2

x

2x + 2

Determine the value of x in the following right-angled triangles:

Exercise 3A1 2 365

4 987

+

-x

_..%

c

.0 =

18Kickass Maths - The theorem of Pythagoras

#1 #2P

Q R M

T

K

x + 2

x + 1

�� 12

52

x

132

x

Calculate the numerical value of the lateral lengths in the following right-angledtriangles:

Exercise 3B1 2 365

4 987

+

-x

_..%

c

.0 =

Documents

Gr 11 Kickass Maths final · 2020-07-24 · 1 Kickass Maths - Introduction and examination guidelines You must know ALL theorems and geometry facts. You must be able to apply ALL