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Euclidian Geometry
Grade 10 to 12
(CAPS)
Compiled by Marlene Malan
Prepared by
Marlene Malan
CAPS DOCUMENT (Paper 2)
Grade 10 Grade 11 Grade 12 (a) Revise basic results
established in earlier
grades.
(b) Investigate line segments
joining the midpoints of two
sides of a triangle.
(c) Properties of special
quadrilaterals.
(a) Investigate and prove theorems of
the geometry of circles assuming
results from earlier grades, together
with one other result concerning
tangents and radii of circles.
(b) Solve circle geometry problems,
providing reasons for statements
when required.
(c) Prove riders.
(a) Revise earlier (Grade 9) work on the necessary
and sufficient conditions for polygons to be similar.
(b) Prove (accepting results established in earlier
grades):
• that a line drawn parallel to one side of a triangle
divides the other two sides proportionally (and the
Mid-point Theorem as a special case of this
theorem);
• that equiangular triangles are similar;
• that triangles with sides in proportion are similar;
• the Pythagorean Theorem by similar triangles;
• riders.
REVISION FROM EARLIER GRADES
SIMILARITY
AAA
or
∠∠∠
SSS
CONGRUENCY
SSS
AAS
SAS
(included angle)
RHS
PROPERTIES OF SPECIAL QUADRILATERALS
PARALLELOGRAM
• Both pairs of opposite sides are parallel
• Both pairs of opposite side are equal
• Both pairs of opposite angles are equal
• Diagonals bisect each other
RECTANGLE
All properties of parallelogram PLUS:
• Both diagonals are equal in length
• All interior angles are equal to 90°
RHOMBUS
All properties of parallelogram PLUS:
• All sides are equal
• Diagonals bisect each other perpendicularly
• Diagonals bisect interior angles
SQUARE
All properties of a rhombus PLUS:
• All interior angles are 90°
• Diagonals are equal in length
KITE
• Two pairs of adjacent sides are equal
• Diagonal between equal sides bisects other
diagonal
• One pair of opposite angles are equal
(unequal sides)
• Diagonal between equal sides bisects
interior angles (is axis of symmetry)
• Diagonals intersect perpendicularly
TRAPEZIUM
• One pair of opposite sides are parallel
HOW TO PROVE THAT A QUADRILATERAL IS A PARALLELOGRAM
Prove any ONE of the following:
• Prove that both pairs of opposite sides are parallel
• Prove that both pairs of opposite sides are equal
• Prove that both pairs of opposite angles are equal
• Prove that the diagonals bisect each other
• Prove that ONE pair of sides are equal and parallel
HOW TO PROVE THAT A PARALLLELOGRAM IS A RHOMBUS
Prove ONE of the following:
• Prove that the diagonals bisect each other perpendicularly
• Prove that any two adjacent sides are equal in length
TRIANGLES BETWEEN PARALLEL LINES
The AREA of two triangles on the SAME (OR EQUAL) BASE between two parallel lines, are EQUAL.
Area of ∆�� = Area of ∆���
MIDPOINT THEOREM
The line segment joining the midpoints of two sides of a triangle, is parallel to the third side
of the triangle and half the length of that side.
( Midpt Theorem )
If AD = DB and AE = EC, then DE ǁ BC and DE =�
BC
CONVERSE OF MIDPOINT THEOREM
If a line is drawn from the midpoint of one side of a triangle parallel to another side, that line will bisect
the third side and will be half the length of the side it is parallel to.
( line through midpoint ⃦ to 2nd side )
If AD = DB and DE ǁ BC, then AE = EC and DE =�
BC.
GRADE 11 GEOMETRY
Note: THEOREMS OF WHICH PROOFS ARE EXAMINABLE ARE INDICATED WITH
Theorem 1 Converse of Theorem 1
If AC = CB in circle O, then OC AB. If OC chord AB , then AC = BC .
(line from centre to midpt of chord) (line from centre to chord)
Theorem 2
The angle at the centre of a circle subtended by an arc/a chord is double the angle at the circumference
subtended by the same arc/chord. AO�B = 2 � AC�B
( ∠ at centre = 2 ×∠ at circumference )
Theorem 3 Converse of Theorem 3
The angle on the circumference subtended by If � = 90°, then AB is the diameter
the diameter, is a right angle. of the circle.
The angle in a semi-circle is 90°.
(∠s in semi circle OR (chord subtends 90° OR
diameter subtends right angle) converse ∠s in semi circle)
Theorem 4 Converse of Theorem 4
The angles on the circumference of a If a line segment subtends equal angles
circle subtended by the same arc or at two other points, then these four points
chord, are equal. lie on the circumference of a circle.
(∠s in the same seg) (line subtends equal ∠s OR
converse ∠s in the same seg)
Corollary of Theorem 4
Equal chords subtend equal Equal chords subtend equal angles Equal chords of equal circles
angles at the circumference at the centre of the circle. subtend equal angles at the
of the circle. circumference.
(equal chords; equal ∠s) (equal chords; equal ∠s) (equal circles; equal chords; equal
∠s)
Theorem 5 Converse of Theorem 5
The opposite angles of a cyclic quadrilateral If the opposite angles of a quadrilateral
are supplementary. are supplementary, then it is a cyclic
�� � � = 180° quadrilateral.
�� � �� 180°
(opp ∠s of cyclic quad ) ( opp ∠s quad sup OR
converse opp ∠s of cyclic quad )
Theorem 6 Converse of Theorem 6
The exterior angle of a cyclic quadrilateral If the exterior angle of a quadrilateral is equal
is equal to the opposite interior angle. to the opposite interior angle, then it is a
cyclic quadrilateral.
(ext ∠ of cyclic quad ) (ext ∠ = int opp ∠ OR converse ext ∠ of cyclic quad)
Theorem 7 Converse of Theorem 7
The tangent to a circle is perpendicular to the If a line is drawn perpendicularly to the radius
radius at the point of tangency. through the point where the radius meets the
circle, then this line is a tangent to the circle.
( tan ⊥ radius OR ( line ⊥ radius OR converse tan ⊥ radius OR
tan ⊥ diameter ) converse tan ⊥ diameter )
Theorem 8
If two tangents are drawn from the same point outside a circle, then they are equal in length.
(tans from common pt OR Tans from same pt )
Theorem 9 (Tan chord theorem) Converse of Theorem 9
The angle between the tangent to a circle If a line is drawn through the endpoint of
and a chord drawn from the point of tangency, a chord to form an angle which is equal to
is equal to the angle in the opposite circle the angle in the opposite segment, then this
segment. line is a tangent.
( tan chord theorem ) ( converse tan chord theorem OR
∠ between line and chord )
Acute angle Obtuse angle
THREE WAYS TO PROVE THAT A QUADRILATERAL IS A CYCLIC QUADRILATERAL
Prove that :
• one pair of opposite angles are supplementary
• the exterior angle is equal to the opposite interior angle
• two angles subtended by a line segment at two other vertices of the quadrilateral, are equal.
GRADE 12 GEOMETRY
The Concept of Proportionality (Revision)
A 6 cm B 4 cm C
D 9 cm E 6 cm F
AB : BC = 6 : 4 = 3 : 2 and DE : EF = 9 : 6 = 3 : 2
Although, AB : BC = DE : EF it does NOT mean that AB = DE, AC = DF or BC = EF.
Theorem 1 Converse of Theorem 1
A line drawn parallel to one side of a triangle If a line divides two sides of a triangle propor-
that intersects the other two sides, will divide tionally, then the line is parallel to the third
the other two sides proportionally. side of the triangle.
( line || one side of Δ ( line divides two sides of Δ in prop )
OR prop theorem; name || lines )
If DE ǁ BC then ����
= � !
or AD : DB = AE : EC If ����
= � !
then DE ǁ BC.
Theorem 2 (Midpoint Theorem) Converse of Theorem 2
(Special case of Theorem 1)
The line segment joining the midpoints of two If a line is drawn from the midpoint of one
sides of a triangle, is parallel to the third side side of a triangle parallel to another side, that
of the triangle and half the length of that side. line will bisect the third side and will be half
the length of the side it is parallel to.
( midpt theorem ) ( line through midpt || to 2nd side )
If AD = DB and AE = EC, then DE ǁ BC and DE =�
BC If AD = DB and DE ǁ BC, then AE = EC and DE =
�
BC.
Theorem 3 Converse of Theorem 3
The corresponding sides of two equiangular If the sides of two triangles are proportional,
triangles are proportional and consequently then the triangles are equiangular and
the triangles are similar. consequently the triangles are similar.
( ||| Δs OR equiangular Δs ) ( Sides of Δ in prop )
If ∆�� ||| ∆�"# then ��
�
�!
$
�!
�$ If
��
�
�!
$
�!
�$ then ∆�� |||∆�"#
Theorem 4
The perpendicular drawn from the vertex of the right angle of a right-angled triangle, divides the triangle in
two triangles which are similar to each other and similar to the original triangle.
Corollaries of Theorem 4
∆��|||∆��� ∆��|||∆�� ∆���|||∆��
∴��
��
�!
��
�!
�� ∴
��
��
�!
�!
�!
�! ∴
��
��
��
�!
��
�!
∴ �' '�.' ∴ �) )�. )� ∴ �* �*.*
Theorem 5 (The Theorem of Pythagoras)
From the corollaries it can be proven that: � �� � �
TIPS TO SOLVING GEOMETRY RIDERS
• READ-READ-READ the information next to the diagram thoroughly
• TRANSFER all given information to the DIAGRAM
• Look for KEYWORDS, e.g.
TANGENT: What do the theorems say about tangents?
CYCLIC QUADRILATERAL: What are the properties of a cyclic quad?
• NEVER ASSUME something!
- Don’t assume that a certain line is the DIAMETER of a circle unless it is clearly
state or unless you can prove it
- Don’t assume that a point is the CENTRE of a circle unless it is clearly stated
(“circle M” means “the circle with midpoint M”)
• Set yourself “SECONDARY” GOALS, e.g.
- To prove that �� = � (primary goal), first prove that
�� = � (secondary goal) and vice versa
- To prove that line AC is a tangent (primary goal), first prove that the line is
perpendicular to radius OB (secondary goal)
AC is tangent
- To prove that BC is the diameter of the circle (primary goal), first prove that �� =90° (secondary goal)
BC is the diameter of the circle
• For questions like: Prove that ��� � . Start with ONE PART.
Move to the OTHER PART step-by-step stating reasons.
Remember it has to be clear and logical to the reader!
E.g. ��� = �� ; �� = �� ; �� = � ; ∴ ��� = �
GRADE 11 GEOMETRY SAMPLE QUESTIONS
Question 1
AB and CD are two chords of the circle with centre O.
+"⏊� , AF = FB, OE = 4 cm, OF = 3 cm and AB = 8 cm.
Calculate the length of CD. [8]
Question 2
O is the centre of the circle. STU is a tangent at T.
BC = CT
�-� 105° and -�/ = 40°
Calculate, giving reasons, the size of:
2.1 �� (2)
2.2 ��� (2)
2.3 �� (3)
2.4 � (6)
[13]
Question 3
3.1 Write down with reasons four other angles
which are equal to 1. (8)
3.2 Prove that ∆ABC||| ∆EDC. (4)
3.3 Prove that � = �!. !
�! (2)
[14]
Question 4
O is the centre of the circle. BC = CD
Express the following in terms of 1:
4.1 �� (2)
4.2 ��� (3)
4.3 �� (4)
[9]
Question 5
LOM is the diameter of circle LMT. The centre of
the circle is O. TN is a tangent at T.
23⏊34
Prove that:
5.1 MNPT is a cyclic quadrilateral. (3)
5.2 NP = NT (6)
[9]
Question 6
PA and PC are tangents to the circle at C and A.
AD ǁ PC and PD intersects the circle at B.
Prove that:
6.1 � bisects 4��� (6)
6.2 ��� ��5 (6)
6.3 �4� ���� (4)
[16]
Question 7
TA is a tangent to the circle. M is the centre of chord PT.
-�⏊4�. O is the centre of the circle.
Prove that:
7.1 MTAR is a cyclic quadrilateral. (3)
7.2 PR = RT (4)
7.3 TR bisects PTA (4)
7.4 -� �
+�� (4)
[15]
GRADE 12 GEOMETRY SAMPLE QUESTIONS
Example
Given: ��:�� 2: 3 and �" 8
5".
Instruction: Determine the ratio of 4: 4�.
Solution:
In ∆��": � 9
= : ∴ '; = :
<"
But it was given that '; = 85
"
∴ 85
" = :
<"
!9
= :
÷ 85
= �:>
In ∆�< : !??�
= ! 9
= �:>
4: 4� = 15: 8
Question 1
"� = 22@A,� 33@A, � 15@ACDE�� 1.
Calculate the value of 1. [4]
Question 2
�#||", � = 5>
�CDE�": "� 4: 3
Determine the ratio �F:F�. [8]
Question 3
G�
GH
�
5, 4�: �I 1:2CDE4J||��.
3.1 Write down the values of I�: I4and I�:�K. (2)
3.2 Determine �J:�I (1)
3.3 Prove that IJ JK. (6)
[9]
Question 4
Given: 4KL|��CDE4I|L��
Prove that KI||��. [4]
Question 5
∆4K-is inscribed in a circle. �+||KI, 4� = �KCDE4� �-
PR is the diameter of the circle.
Prove that:
5.1 ��||K- (2)
5.2 O is the centre of the circle (2)
5.3 BORT is a trapezium. (2)
[6]
Question 6
Given: 4�: �K = 5: 4CDE4�:�I 5: 2
S is the midpoint of AQ
6.1 Prove that �- 2MI (8)
6.2 If I<||KN, determine 4N: N- (6)
[14]
Question 7
Rectangle DEFK is drawn inside right-angled ∆ABC.
Prove that:
7.1 ��. �� = �". �< (4)
7.2 �": " = �": # (4)
7.3 <#: " = �": # (1)
7.4 ����
= �!�
(3)
[12]
Question 8
ABOC is a kite with �� = � = 90°
8.1 Why is ∆+�|||∆+�? (2)
8.2 Complete:
8.2.1 + =. . .� …
8.2.2 � =. . .� …
8.2.3 � =. . .� … (3)
8.3 Prove that ��P
Q�P = ���Q
(3)
8.4 Prove that + − +� = +�. �� (2)
8.5 If +� = �
�� = 1, prove that � = √2. +� (2)
[12]
MIXED EXERCISES
1. In the diagram, TBD is a tangent to circles BAPC and BNKM at B.
AKC is a chord of the larger circle and is also a tangent to the smaller circle at K.
Chords MN and BK intersect at F. PA is produced to D.
BMC, BNA and BFKP are straight lines.
Prove that:
a) MN ǁ CA
b) ∆<J3 is isosceles
c) �99?
= �TT!
d) DA is a tangent to the circle passing through
points A, B and K.
2. In the diagram below, chord BA and tangent TC of circle ABC are produced to meet at R.
BC is produced to P with RC=RP. AP is not a tangent.
Prove that:
a) ACPR is a cyclic quadrilateral.
b) ∆��|||∆I4�
c) I = !�.G��!
d) I�. � = I. �
e) Hence prove that I = I�. I�
3. In the diagram alongside, circles ACBN and AMBD
Intersect at A and B.
CB is a tangent to the larger circle at B.
M is the centre of the smaller circle.
CAD and BND are straight lines.
Let ��5 = 1
a) Determine the size of �� in terms of 1.
b) Prove that:
i) CB ǁ AN
ii) AB is a tangent to circle ADN.
4. In the diagram below, O is the centre of circle ABCD.
DC is extended to meet circle BODE at point E.
OE cuts BC at F. Let"�� = 1.
a) Determine �� in terms of 1.
b) Prove that:
i) BE=EC
ii) BE is NOT a tangent
to circle ABCD.
5. In the diagram alongside, medians AM and CN of ∆�� intersect at O.
BO is produced to meet AC at P.
MP and CN intersect in D.
ORǁMP with R on AC.
a) Calculate, giving reasons, the numerical value of U�U!
.
b) Use �+: �J = 2: 3, to calculate the numerical value of G?
?!.
6. In the diagram, AD is the diameter of circle ABCD.
AD is extended to meet tangent NCP in P.
Straight line NB is extended to Q and intersect AC in M with Q on straight line ADP.
AC ⏊NQ at M.
a) Prove that NQ ǁ CD.
b) Prove that ANCQ is a cyclic quadrilateral.
c) i) Prove that ∆4�|||∆4�.
ii) Hence, complete: 4 = ⋯
d) Prove that � = �. 3�
e) If it is further given that PC=MC, prove that
1 − �TP
�!P = �?.�?!�.U�
SOLUTIONS TO MIXED EXERCISE
1. a) ��� = J�� tan chord
��� = �
∴ J�� = �
∴ J3||� corr ∠s =
b) <�� = J� alt ∠s
<�� = 3� tan chord
∴ ∆<J3 is isosceles
c ) <�8 = 3� alt ∠s
3� = ��5 ∠s in same segment
��5 = ��5 ∠s in same segment
∴ <�8 = ��5
∴ 3<||�4 alt ∠s=
∴ �UU�
= �99?
line || to one side of ∆
But �UU�
= �TT!
line || to one side of ∆
∴ �99?
= �TT!
d) ��5 = ��5 ∠s in same segment
��5 = �� equal chords subt equal ∠s
∴ ��5 = ��
∴ �� is a tangent to the circle through A, B and K
2. a) �5 = 4�I ∠s opp equal sides
�5 � � = ��� � �� ext ∠ of ∆
� = �� tan chord
∴ �5 = ���
∴ ��� = 4�I both = �5
ACPR is a cyclic quadrilateral (ext ∠of quad)
b) In ∆�� and ∆I4�:
4� = � ∠s in same segment
= �� proven in 2 a
∴ �� = 4�
�� = �I�4 ext ∠of cyclic quad
��� ��5 3rd ∠ of∆ ∴ ∆��|||∆I4�∠∠∠
c) G?
!�
G�
!� from 2 b
I4 !�.G�
!� but I4 = I
∴ I = !�.G�!�
d) In ∆I� and ∆I�:
� = �� tan chord
I�� is common
I�� = I�� 3rd angle
∴ ∆I�|||∆I�∠∠∠
�!
!�
G!
G� ∆W|||
I�. � = I. �
e) !�G?
= !�G�
from 2. b)
!�G!
= !�G�
RC=RP
� = !�.G�G!
From 2.d) � = G!.!�G�
∴ !�.G�G!
= G!.!�G�
∴ I = I�. I�
3. a) �� = ��5 = 1 ∠s opp equal sides
J�� = 180° − 21 sum ∠s of ∆
∴ �� = 21
b. i) � T�X
∠ at centre =2x∠circ
90° − 1 ��� 180° − (90° − 1 � 21Y sum ∠s of ∆ 90° − 1 3�� � 90° − 1 ext ∠of cyclic quad
∴ ��� 3��
∴ �||�3 corr ∠s
b. ii) ��� = �� = 21 tan chord
��� �� alt ∠s
�� ��
∴AB is a tangent ∠betw line&chord
4. a) ��5 "�� 1 ∠s in same segment
��5 �� 1 ∠s opp = sides
�+�� 180° − 21 sum ∠s of ∆
�� = 90° − 1 ∠ at centre =2x∠atcircumference
b. i) �� 90° − 1 ext ∠of cyclic quad
#� 180° − (1 � 90° − 1Y sum ∠s of ∆
= 90°
In ∆�"# and ∆"#:
#�� = #� = 90° ∠sonstr line
BF = FC
FE is common
∆�"# ≡ ∆"# s∠s
BE = EC ∆W ≡
b. ii) ��� = 90° − 1 sum ∠s of ∆
∴ ��� = ��
∴ BE is not a tangent `��� � �� ≠ ��b
5. a) P is midpoint of AC medians concur
AB||PM midpt theorem
In ∆�3:
U�U!
= �T�!
= �?�!
line || one side of ∆
= �T �T
= �
b) In ∆�J4:
�QQT
= QTQT
G??!
= G?�?
BP is a median
= QT�T
line || one side of ∆
= QT5QT
= �5
6. a) � = 90° ∠ in semi ⊙
J� = 90° AM⏊NM
∴ 3K||� corr ∠s=
b) �� = 3� || lines, corr ∠s
�� = �� tan chord
= 3�
∴ ANCQ is a cyclic quad ∠s subt by same line segm
c) i) In ∆4� and ∆4�:
�� = �� tan chord
4� is common
��� = ��4 3rd ∠
∴ ∆4� ||| ∆4� ∠∠∠
c) ii) 4 = �4. �4
d) In ∆3� and ∆��:
3� = �� ∠s in same segm
= �� ∠s in same segm
�8 = ��� tan chord
= �� ∠s in same segm
��� = ��� 3rd ∠ ∴ ∆3� ≡ ∆�� ∠∠∠
∴�!
U�
!�
U�
� �. 3�
eY 1 − �TP
�!P = �!Pe�TP
�!P
T!P
�!P Pyth.
?!P
�!P
�?.�?!�.U�