gr 11 kickass maths final · 2020-07-24 · 1 kickass maths - introduction and examination...
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KickassMathsKicckassMathsKassMathKickasssMathKickasMathKickassMatathKickassMaKicassMathKic-assMathKic-kassMathKickass-sMathKickaMathKickassMath-KickassMaKickassMathKick-sMathKicassMathKickasKickas
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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:
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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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
_..%
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 �
8Kick ass Maths - Revision of previous geometric concepts
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 �������������������
+ = + +
= +
A Kick ass Maths - Revision of previous geometric concepts9
Parallel lines
F������%��������������
>
> =
U��&����������������
> >
+ = 180°
N�������������������������
> >
=
10Kick ass Maths - Revision of previous geometric concepts
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� :��������
A Kick ass Maths - Revision of previous geometric concepts11
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
12Kick ass Maths - Revision of previous geometric concepts
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
A Kick ass Maths - Revision of previous geometric concepts13
#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 =
14Kick ass Maths - Revision of previous geometric concepts
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 =