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EUCLIDEAN
GEOMETRY
GRADE 12
NOTES AND ACTIVITIES
EMAILBY
ADDRESS:
melulekishabalala@gmail.com
CELLPHONE NUMBER: 0733318802 Page 1
MR M. SHABALALA
@NOMBUSO HIGH
EUCLIDEAN GEOMETRY GRADE 10-12
EUCLIDEAN
GEOMETRY
HAS 3 BRANTCHES
1.LINES
2. TRIANGLES/ QUADRILATERALS
3.CIRCLES
S
SUMMARY OF EUCLIDEAN GEOMETRY FROM EARLY GRADES


START? Draw any two lines that will be asked in exam OR
REVISION? List 5 facts about parallel lines
1. FACTS ABOUT THE LINES
1.1. PERPENDICULAR LINES
OR
 Cˆ1  Cˆ 2  180 (∠′𝒔 on str line)
 BUT  AD   ( AC )2   CD  (Pythagoras)
2
2
 ABC  ADC (RHS)
1.2. PARALLEL LINES
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 2
 Alternating angles
Eˆ3  Fˆ2 (Alt ∠’s AB // CD)
Eˆ  Fˆ (Alt ∠’s AB // CD)
4
3
 Corresponding angles
Eˆ1  Fˆ2 (Corresp ∠’s AB // CD)
Fˆ  Eˆ (Corresp ∠’s AB // CD)
4
3
Eˆ 4  Fˆ1 (Corresp ∠’s AB // CD)
 Co-interior angles
Eˆ 4  Fˆ2  180 (Co-int ∠’s AB // CD)
Fˆ  Eˆ  180 (Co-int ∠’s AB // CD)
3
2
1.3. CROSSING LINES
 Eˆ1  Eˆ3 (Vert opp ∠’s )
 Eˆ 2  Eˆ 4 (Vert opp ∠’s )
 Eˆ1  Eˆ 2  Eˆ3  Eˆ 4  360 (∠’s around pt)
2. FACTS ABOUT THE TRIANGLE
2.1 Properties of triangles must be known (isosceles, equilateral, right angled triangle etc.)
(a) Aˆ  Bˆ  Cˆ1  180 (Sum of ∠′𝒔 in ∆)
(b) Aˆ  Bˆ  Cˆ 2 (ext ∠ of ∆)
(c) B̂  Cˆ1 (opp ∠′𝒔 = sides )
EXAMPLE
Solve for the unknown angles in each of the following triangles.
180  106
1.Bˆ  Cˆ 
 37 (opp ∠′𝒔 = sides)
2
BY MR M. SHABALALA
2. Aˆ  40 (ext ∠ of ∆) , Cˆ 2  30
@NOMBUSO HIGH
Page 3
2.2 CONGRUENT TRIANGLES

Congruent triangles – are triangles that are exactly the same in every aspect, equal
in shape and size
2.3 SIMILAR TRIANGLES
 Similar triangles – Triangles that are congruent or are triangles that have same shape
but different in size. If two triangles are equiangular, then their corresponding sides
are in proportion vice versa. When naming similar triangles order proportion is very
important.
In ΔABC and ΔDEF
1. Â = D̂ (Given)
2. B̂ = Ê (Given)
3. Ĉ = F̂ (sum of ∠ ‘s of a Δ)
∴ ΔABC ∣∣∣ ΔDEF (∠∠∠ ) or (AAA) or (Equiangular Δ′s)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 4
3. SUMMARY OF THE THEOREMS ABOUT CIRCLES

Circle theorems are divided into four groups
GROUP 1 (CENTRE GROUP)
1.( Line from centre ⊥ chord)
2. ( ∠ at the centre = 2× ∠ at the circumference)
GROUP 2 (NON CENTRE GROUP)
1.( ∠’s in same seg)
2.( equal chords, equal angles)
THIS DOES NOT EXIST
GROUP 3 (CYCLIC QUADRILATERAL GROUP
̂ = 𝟏𝟖𝟎°
̂+𝑪
1. 𝑨
′
(𝒐𝐩𝐩 ∠ 𝐬 𝐨𝐟 𝐜𝐲𝐜𝐥𝐢𝐜 𝐪𝐮𝐚𝐝 )
̂𝑬
̂ = 𝑫𝑪
2. 𝑨
( ext ∠ in cyclic quad)
3. ( ∠’s in same seg)
GROUP 4 (TANGENT GROUP)
𝟏. (𝐭𝐚𝐧 ⊥ 𝐫𝐚𝐝𝐢𝐮𝐬 )
BY MR M. SHABALALA
𝟐. (𝐭𝐚𝐧 𝐜𝐡𝐨𝐫𝐝 𝐭𝐡𝐞𝐨𝐫𝐞𝐦 )
@NOMBUSO HIGH
𝟑. (𝐭𝐚𝐧𝐬 𝐟𝐫𝐨𝐦 𝐜𝐨𝐦𝐦𝐨𝐧 𝐩𝐭 )
Page 5
FROM GRADE 11’S PROOF TO BE KNOWN FOR EXAM PURPOSES
3.1 PROOF 1 (LINE FROM THE CENTRE)
O is the centre of the circle and AB is a chord. OB ⊥ AC. Prove the
theorem which states that AB = BC
(99(
(5)
GIVEN
R.T.P
CONSTRUCTION
PROOF:
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 6
3.2 PROOF 2(ANGLE AT THE CENTRE)
In the figure Below o is the centre of the circle and A,B and C are three
points on the circumference of the circle. Use the figure and prove the
theorem that states that.
𝐵𝑂̂𝐶 = 2𝐴̂
(5)
GIVEN
R.T.P
CONSTRUCTION
PROOF:
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 7
3.3 PROOF 3(CYCLIC QUAD PROOF)
In the diagram below, O is the centre of the circle. Use the diagram to prove the
theorem which states that: If PQRS is a cyclic quadrilateral then 𝑃̂ + 𝑅̂ = 180° (5)
GIVEN: _______________________________
R.T.P
:_______________________________
CONSRUCTION: _______________________
PROOF:
3.4 PROOF 4( TANGENT PROOF)
In the diagram below the circle with centre O passes through points T,
R, and S. QS is a tangent to the circle at S. TS, TR and SR are joined.
Prove that 𝑄𝑆̂𝑇= 𝑅̂
(6)
GIVEN: _______________________________
R.T.P
:_______________________________
CONSRUCTION: _______________________
PROOF:
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 8
REVISION ACTIVITIES EARLY GRADES TO GRADE 11
QUESTION 1
1.1
In the diagram O is the centre of a circle which passes through P, Q, R and T. QT, OQ
and OT are joined. OT is parallel to QR. 𝑄̂2 = 32°, 𝑃̂ = 𝑥 and 𝑇̂3 = 𝑦.
Determine, with reasons, the size of 𝑥 and 𝑦.
1.2
(8)
In the diagram A, B,C and D are points on the circumference of the circle with centre O.
AOC is a diameter. D𝐴̂𝐶 = 22°
Calculate, with reasons, the size of 𝐵̂.
BY MR M. SHABALALA
@NOMBUSO HIGH
(5)
Page 9
QUESTION 2
2.1
Complete the following statement:
The angle between the tangent and the chord is equal ...
2.2
(1)
In the diagram points P, Q, R and T lie on the circumference of a circle. MW and
TW are tangents to the circle at P and T respectively. PT is produced to meet RU
at U.𝑀𝑃̂𝑅 = 75° ,
𝑃𝑄̂ 𝑇 = 29° ,
𝑄𝑇̂𝑅 = 34°
Let 𝑇𝑃̂𝑊 = 𝑎, 𝑅𝑃̂ 𝑇 = 𝑏, 𝑀𝑃̂𝑄 = 𝑐 and 𝑅𝑇̂𝑈 = 𝑑, calculate the values of 𝑎, 𝑏, 𝑐 and 𝑑.
[9]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 10
QUESTION 3
3.1
In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle. AT
produced and CK produced meet in N. Also NA = NC and 𝐵̂ = 38°
3.1.1
Calculate, with reasons, the size of the following angles:
(a)
̂A
KM
(2)
(b)
̂2
T
(2)
(c)
Ĉ
(2)
(d)
̂4
K
(2)
3.1.2
Show that NK = NT.
(2)
3.1.3
Prove that AMKN is a cyclic quadrilateral.
(3)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 11
GRADE 11 REVISION ACTIVITIES ABOUT PROVE
HINT: Prove that?
1. 1st ask yourself, what if, if it is like that?
2. Then reverse, and prove.
QUESTION 4
ALB is a tangent to circle LMNP. ALB ∥ MP.
Prove that:
4.1
LM = LP
(4)
4.2
̂P
LN bisects MN
(4)
4.3
LM is a tangent to circle MNQ
(4)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 12
QUESTION 5
5.1
In the diagram O is the centre of the circle. KM and LM are tangents to the
circle at K and L respectively. T is a point on the circumference of the circle.
KT and TL are joined. 𝑂̂1 = 106°.
5.1.1 Calculate, with reasons, the size of 𝑇̂1 .
(3)
5.1.2 Prove that quadrilateral OKML is a kite.
(3)
5.1.3 Prove that quadrilateral OKML is a cyclic quadrilateral.
(3)
̂.
5.1.4 Calculate, with reasons, the size of 𝑀
(2)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 13
QUESTION 6 (2015)
In the diagram below, ΔABC is drawn in the circle. TA and TB are tangents to the circle. The
straight line THK is parallel to AC with H on BA and K on BC. AK is drawn.
Let 𝐴̂3 = 𝑥
6.1
̂3 = 𝑥
Prove that 𝐾
(4)
6.2
Prove that AKBT is a cyclic quadrilateral.
(2)
6.3
̂ 𝐵.
Prove that TK bisects 𝐴𝐾
(4)
6.4
Prove that TA is tangent to the circle passing through the points A, K and H.
(2)
6.5
S is a point in the circle such that the points A, S, K and B are concyclic.
Explain why A, S, B and T are also concyclic.
BY MR M. SHABALALA
@NOMBUSO HIGH
(2)
Page 14
QUESTION 7 (2015)
In the diagram, the vertices A, B and C of ∆𝐴𝐵𝐶 are concyclic. EB and EC are
tangents to the circle at B and C respectively. T is a point on AB such that
TE∥ AC. BC cuts TE in F.
7.1
Prove that 𝐵̂1 = 𝑇̂3
(4)
7.2
Prove the TBEC is a cyclic quadrilateral.
(4)
7.3
̂C.
Prove that ET bisects BT
(2)
7.4
If it is given that TB is a tangent to the circle through B, F and E,
7.5
prove that TB = TC.
(4)
Hence, prove that T is the centre of the circle through A, B and C,
(3)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 15
QUESTION 8 (2013)
8.1
Complete the following statement so that it is valid:
The angle between a chord and a tangent at the point of contact is … (1)
8.2
In the diagram, EA is a tangent to circle ABCD at A.
AC is a tangent to circle CDFG at C. CE and AG intersect in D.
If 𝐴̂1 = 𝒙 and
𝐸̂1 = 𝒚, prove the following with reasons:
8.2.1 BCG ∥ AE
(5)
8.2.2 AE is a tangent to circle FED
(5)
8.2.3 AB = AC
(4)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 16
IN GRADE 12 THERE ARE 2 NEW THEOREMS
1.
PROPORTIONALITY THEOREM states that, a
line drawn parallel to one side of a triangle divides the other two sides proportionally.
 REASON ( Prop theorem, name ∥ lines or line ∥ one side of ∆)
AD
DB
=

AE
EC
( Prop theorem, DE ∥ BC ) OR
AB
BD
=
AC
EC
OR
………
……..
=
……….
……….
As a ratio: AD:DB = AE:EC
PROOF FOR PROPORTIONALITY THEOREM

Before the proof remember the following:
1. Parallel Lines, corresponding angles
2. Triangles with equal or common bases, lying between the same parallel lines are all have equal area or have the same area.
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 17
± 7 Marks
PROOF - PROPORTIONALITY THEOREM
1. A line drawn parallel to one side of a triangle divides the other two sides
Proportionally
reason (line || one side of Δ or prop theorem; name || lines)
GIVEN
:
Any ΔABC with DE // BC, D on AB ( or its extension ) and E
on AC (or its extension).
R.T.P
:
Prove that
𝐴𝐷
𝐷𝐵
=
𝐴𝐸
𝐸𝐶
CONSTRUCTION : Draw altitude h ⊥ to base AD and altitude k ⊥ to base

AE. Join DC and BE
PROOF:
𝐴𝑟𝑒𝑎 ∆𝐴𝐷𝐸
𝐴𝑟𝑒𝑎 ∆𝐵𝐷𝐸
𝐴𝑟𝑒𝑎 ∆𝐴𝐸𝐷
𝐴𝑟𝑒𝑎 ∆𝐶𝐸𝐷
=
1
. 𝐴𝐷.ℎ
2
1
. 𝐵𝐷.ℎ
2
=
=
1
. 𝐴𝐸.𝑘
2
1
. 𝐸𝐶.𝑘
2
=
𝐴𝐷
𝐷𝐵
𝐴𝐸
𝐸𝐶
But 𝐴𝑟𝑒𝑎 ∆𝐵𝐷𝐸 = 𝐴𝑟𝑒𝑎 ∆𝐶𝐸𝐷
∴
𝐴𝑟𝑒𝑎 ∆𝐴𝐷𝐸
𝐴𝑟𝑒𝑎 ∆𝐵𝐷𝐸
𝐴𝐷
𝐷𝐵
=
=
(same height h)
S 𝑅
(same height k)
S/R
(Same base between DE//BC) S 𝑅
𝐴𝑟𝑒𝑎 ∆𝐴𝐸𝐷
𝐴𝑟𝑒𝑎 ∆𝐶𝐸𝐷
𝐴𝐸
S
(7)
𝐸𝐶
Similarly
𝐴𝐷
𝐷𝐵
=
𝐴𝐸
𝐸𝐶
𝑎𝑛𝑑
BY MR M. SHABALALA
𝐴𝐵
𝐵𝐷
=
𝐴𝐶
𝐶𝐸
@NOMBUSO HIGH
Page 18
ACTIVITIES FOR PROPORTIONALITY THEOREM AND HINTS
 Identify parallel lines, and use ratios for proportion.
 Useful strategies in solving problems with proportion involving areas of triangles:
 Use area rule if there is a common angle in triangles in the question
𝟏
Area = 𝟐 𝒂. 𝒃𝒔𝒊𝒏𝑪
𝟏
 Or use Area = 𝟐 𝒃. 𝒉 if there is a common vertex or same height or same base in
triangles in question)
 If none of the above, Identify a common triangle and relate the two triangles in
question to it, then use any of the two methods mentioned above.
QUESTION 1
In the figure below DE || FG || BC.
AD = 36 cm, DF = 24 cm, AE = 48 cm and DE = GC = 40 cm.
Determine, with reasons, the lengths of:
1.1
EG
(2)
1.2
BC
(4)
QUESTION 2
In the diagram below, ΔVRK has P on VR and T on VK such that PT || RK. VT = 4 units,
PR = 9 units, TK = 6 units and VP = 2𝑥 – 10 units.
Calculate the value of 𝑥.
(4)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 19
QUESTION 3(JIT)
In the diagram below, P is the midpoint of AC in ∆ABC. R is a point on AB such
that RS ∥ BP and
AR
AB
3
= . RC intersects BP in T.
5
Determine with reasons, the following ratios:
3.1
3.2
3.3
3.4
AS
(4)
SC
RT
(3)
TC
Area of ∆RAS
(2)
Area of ∆RSC
Area of ∆TPC
(3)
Area of ∆RSC
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 20
QUESTION 4(JENN)
In ∆DEF, GH ∥ EF and KH ∥ GF. DK = 80 units and KG = 120 units.
Determine, giving reasons.
4.1
4.2
4.3
DH
HF
in simplest fraction form.
The length of DE.
Area of ∆DHK
Area of ∆DGF
QUESTION 5(FS2020)
5.1 Complete the following statement.
If two triangles are equiangular then, the corresponding sides are …
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 21
5.2
Use the diagram below to prove the theorem which states that a line drawn parallel to one
side of a triangle divides the other two sides proportionally, that is prove that
𝑋𝐾
𝐾𝑌
=
𝑋𝐿
𝐿𝑍
(6)
5.3
In the figure, D is a point on side BC of ∆ABC such that BD = 6 cm and DC = 9 cm.
T and E are points on AC and DC respectively such that TE ∥ AD and AT : TC = 2: 1
5.3.1
Show that D is the midpoint of BE
(3)
5.3.2
If FD = 2 cm , calculate the length of TE.
(3)
5.3.3
Calculate the numerical value of :
a)
b)
Area of ∆ADC
Area of ∆ABD
Area of ∆TEC
Area of ∆ABC
BY MR M. SHABALALA
@NOMBUSO HIGH
(3)
(3)
Page 22
QUESTION 6(WC2019)
In ∆ABC in the diagram, D is a point on AB such that AD : DB = 5 : 4. P and E are points on
AC such that DE ∥ BC and DP ∥ BE. BC is NOT a diameter of the circle.
̂ E = 120°, EC= 12 units and BC = 27 units.
Given: 𝐵𝐷
6.1 Determine with reasons:
6.1.1
6.1.2
6.2
The length of AE.
(3)
Area of ∆AEB
(2)
Area of ∆ECB
Hence, determine the length of DP if ∆ADP ||| ∆ABE
(6)
(After Similarity)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 23
QUESTION 7
7.1
In the figure, DE ∥ BH and FG ∥ EC, where F is the midpoint of line segment AE. It is
further given that
7.1.1
7.1.2
7.1.3
7.1.4
7.2
AD
DB
2
= . Determine:
3
DE
(3)
BH
AG
(1)
AC
FE
(2)
EH
Area of ∆ADE
(3)
Area of ∆ABH
In the figure, AF = 2CG and FE ∥ GB.
7.2.1
Determine
7.2.2
Determine
7.2.3
Determine
AE
AB
=
2
5
AF
(3)
FG
CH
(4)
HE
Area of ∆BCG
Area of ∆AFE
BY MR M. SHABALALA
case3
@NOMBUSO HIGH
(4)
Page 24
QUESTION 8
In the figure below, ∆ABC has DE ∥ BC. AC = 20 cm, EC = 4 cm and DB = 2 cm.
𝐸̂1 = 𝐸̂2 = 𝑥
8.1
Calculate the numerical value of AD.
8.2
Prove that
8.3
Calculate the numerical value of:
AD
DB
Area of ∆ABE
Area of ∆ADE
=
×
AE
EB
.
(4)
Area of ∆ABC
Area of ∆EBC
BY MR M. SHABALALA
(3)
@NOMBUSO HIGH
(4)
Page 25
2.
SIMILARITY THEOREM states that, if two triangles are equiangular, then
their corresponding sides are in proportion..
 REASON ( ||| ∆′𝑠 or equiangular ∆′𝑠 )
 Order is very important when naming similar triangles.
 Note: If two triangles are congruent, then they are also similar, but two similar
triangles are not necessarily congruent.
̂=P
̂ and Ĉ = R
̂ , B
̂=Q
̂
Given: A
 If, ΔABC ||| ΔPQR (AAA)
 Then,
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
=
𝐴𝐶
𝑃𝑅
(||| ∆′𝑠 )
 Example if ΔKLM ||| ΔNOP (AAA)
Write proportion:
PROOF FOR SIMILARITY THEOREM

Before the proof remember the following:
1. Parallel Lines, corresponding angles
2. Cases of congruency
3. Proportionality theorem
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 26
± 7 Marks
PROOF - SIMILARITY THEOREM
2. If two triangles are equiangular(AAA), then the corresponding sides are in
Proportion.
Reason (||| Δs
OR
equiangular Δs)
Given
̂=P
̂ and Ĉ = R
̂ , B
̂=Q
̂
: ΔABC and ΔPQR with A
Prove
:
𝐴𝐵
𝑃𝑄
𝐴𝐶
=
𝑃𝑅
=
𝐵𝐶
𝑄𝑅
Construction: X and Y on AB, AC respectively such that AX = PQ
 all 3 sides
and AY = PR. Join XY.
: In ΔAXY and ΔPQR
Proof
1. AX = PQ (construction)
2. AY = PR (construction)
̂=P
̂
3. A
(given)
∴ ∆𝑨𝑿𝒀 ≡ ∆𝑷𝑸𝑹
(SAS)
Reason
̂
̂𝐘 = 𝑸
∴ 𝐀𝐗
S
(∆′ 𝑠 congruent)
̂
̂=𝑸
But 𝑩
𝐠𝐢𝐯𝐞𝐧
̂𝐘 = 𝑩
̂
∴ 𝐀𝐗
(𝑄̂ = 𝐵̂
∴ XY ∥ BC
( corresponding ∠′ 𝑠 =)
AB
(Prop theorem, XY || BC) S/R
AX
∴
=
𝐴𝐵
𝑃𝑄
AC
AY
=
𝐴𝐶
𝑃𝑅
given)
(AX = PQ and AY = PR)
S
Reason
S
(7)
Similarly it can be proved that ΔABC ||| ΔPQR
∴
𝐴𝐵
𝑃𝑄
=
𝐴𝐶
𝑃𝑅
=
BY MR M. SHABALALA
𝐵𝐶
𝑄𝑅
@NOMBUSO HIGH
Page 27
ACTIVITIES FOR SIMILARITY AND HINTS
 To prove that triangles are similar
 AAA – if all angles are equals it mean triangles are similar
 SSS− if all sides corresponding sides are equal, proportion can be calculated
once it is equal then the triangles are similar
 Possible questions in similarity theorem
1. Prove that triangles are similar e.g ΔABC∣∣∣ΔDEF.
𝐴𝐵
𝐴𝐶
2. Prove that 𝑃𝑄 = 𝑃𝑅. First prove: ΔABC∣∣∣ΔPQR and then deduce the proportion
of the sides.
3. Prove that: KN. PX = NR. YP. Find two triangles in which KN, PX, NR and YP
(or sides equal to these), and thus prove that: ΔKNR∣∣∣ΔYPX , then deduce what
you were asked to prove. USE 1212 METHOD to identify triangles
4. Prove: Proportion with square and + in between, there is a possibility that two
similarities were used or Pythagoras theorem was used.
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 28
QUESTION 1
Two circles touch each other at point A. The smaller circle passes through O, the centre of the
larger circle. Point E is on the circumference of the smaller circle. A, D, B and C are points on
the circumference of the larger circle.
OE || CA.
1.1
Prove, with reasons, that AE = BE.
(2)
1.2
Prove that ΔAED ||| ΔCEB.
(3)
1.3
Hence, or otherwise, show that AE 2 = DE. CE.
(3)
1.4
If AE. EB = EF. EC, show that E is the midpoint of DF.
(3)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 29
QUESTION 2
CD is a tangent to circle ABDEF at D. Chord AB is produced to C. Chord BE cuts chord
̂4 = 𝑥 and 𝐷
̂1 = 𝑦.
AD in H and chord FD in G. AC || FD and FE = AB. Let 𝐷
2.1
Determine THREE other angles that are each equal to 𝑥.
(6)
2.2
Prove that ΔBHD ||| ΔFED.
(5)
2.3
Hence, or otherwise, prove that AB.BD = FD.BH.
(2)
QUESTION 3
In the figure, AD, DC and BE are tangents to the circle.
CO is a radius and chord BC is
drawn. Radius AO is drawn and extended
to cut the circle at J and BE is extended at F.
3.1
Prove that ΔDAH ||| ΔOCH.
3.2
Prove that OH =
3.3
Prove that ΔJBF ||| ΔBAF.
(4)
3.4
Prove that BF 2 = JF. AF
(3)
AO .DH
DC
BY MR M. SHABALALA
(4)
(6)
@NOMBUSO HIGH
Page 30
QUESTION 4
In the diagram, DGFC is a cyclic quadrilateral and AB is a tangent to the circle at B. Chords DB
and BC are drawn. DG and CF produced meet at E and DC is produced to A. EA ∥ GF
4.1
̂1
Give a reason why 𝐵̂1 = 𝐷
(1)
4.2
Prove that ΔABC ||| ΔADB.
(3)
4.3
̂2
Prove that 𝐸̂2 = 𝐷
(4)
4.4
Prove that AE 2 = AD. AC
(4)
4.5
Hence, deduct that AE = AB
(3)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 31
QUESTION 5(WC19)
̂ = 𝑥 and
AP is a tangent to the circle at P. CB ∥ DP and CB = DP. CBA is a straight line. Let 𝐷
𝐶̂2 = 𝑦
Prove with reasons that,
5.1
ΔAPC ||| ΔABP.
(4)
5.2
AP 2 = AB. AC
(2)
5.3
ΔAPC ||| ΔCDP.
(4)
5.4
AP 2 + PC2 = AC2
(4)
QUESTION 6(FS20)
In the diagram, TPR is a triangle with TP = 4,5 units. Point Q and S are on TR and PR
respectively such that QR = 9,6 units, QS = 4 units, TS = 3,6 units, PS = 1,5 units and
SR = 12 units
6.1
6.2
Prove that PT is a tangent to the circle which passes through the points T, S
and R.
Calculate the length of TQ.
BY MR M. SHABALALA
@NOMBUSO HIGH
(7)
(5)
Page 32
QUESTION 7
In the diagram, O is the centre of the circle. PQRS is a cyclic quadrilateral. The tangent through
P intersects RS produced at A. OB⊥ PR and PA ⊥ AS.
Prove that:
7.1
ΔAPS ||| ΔBRS.
(3)
7.2
AP.RS = BR.PS
(2)
7.3
𝑃̂4 = 𝑅̂2
(4)
7.4
BR.RQ = RS.RP
(6)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 33
QUESTION 8
Two circles touch each other internally at B. O, the centre of the bigger circle, lies on the
circumference of the smaller circle. FBG is a common tangent. AB and BC are chords of the
bigger circle and intersect the smaller circle at D and E respectively. AHG forms a straight line.
Chords OD, DE and OB of the smaller circle are drawn. AH:AG = 1:2.
8.1
8.2
Give a reason why:
8.1.1
𝐵̂4 = 𝐻𝐶̂ 𝐵
(1)
8.1.2
̂4
𝐵̂4 = 𝐷
(1)
Prove that:
8.2.1
ΔDBE ||| ΔDBC.
(3)
8.2.2
D is the midpoint of AB
(3)
8.2.3
8.2.4
8.3
AB2
4
= BC. BE
(4)
CE:BC = 5:9 if DB = 3 units and BE = 2 units.
Calculate the ratio of
BY MR M. SHABALALA
Area of ∆AHD
Area of ∆HGB
@NOMBUSO HIGH
(3)
(6)
Page 34
f
EUCLIDEAN
GEOMETRY
GRADE 12
PAST EXAM PAPERS
EMAILBY
ADDRESS:
melulekishabalala@gmail.com
CELLPHONE NUMBER: 0733318802 Page 35
MR M. SHABALALA
@NOMBUSO HIGH
QUESTION 1 (NOVEMBER 2010)
In the diagram below AC is a diameter of the circle with centre O. AC and chord BD intersect
at E. AB, BC and AD are also chords of the circle. OD is joined. AE ⊥ BD.
If 𝐶̂ = 33° , calculate, with reasons, the size of :
1.1
𝐴̂1
(3)
1.2
̂2
𝐷
(2)
1.3
Show that AE bisects 𝐷𝐴̂𝐵
(3)
[8]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 36
QUESTION 2 (NOVEMBER 2010)
2.1
In the diagram below O is the centre of the circle. GH is a tangent to the circle at T.
J and K are points on the circumference of the circle. TJ, TK and JK are joined.
Prove that theorem that states 𝑇̂1 = 𝑇𝐽̂𝐾
2.2
(5)
ED is a diameter of the circle, with centre O. ED is extended to C. CA is a tangent to
the circle at B. AO intersects BE at F. BD || AO. 𝐸̂ = 𝑥 .
2.2.1
Write down, with reasons, THREE other angles equal to 𝑥.
(4)
2.2.2
Determine, with reasons, 𝐶𝐵̂ 𝐸 in terms of 𝑥.
(3)
2.2.3
Prove that F is the midpoint of BE.
(4)
2.2.4
Prove that ΔCBD ||| ΔCEB.
(2)
2.2.4
Prove that 2EF.CB = CE.BD.
(3)
[21]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 37
QUESTION 3 (NOVEMBER 2010)
3.1
In the diagram below A, B, C and D are points on the circumference of the circle.
BD and AC intersect at E. Also,
EB = 8 cm,
DC = 8 cm and
AE : EC = 4 : 7.
If DE = 𝑥 units and AB = 𝑦 units, calculate 𝑥 and y.
[6]
QUESTION 4 (NOVEMBER 2010)
In the diagram below M is the centre of the circle. FEC is a tangent to the circle at E. D is the
midpoint of AB.
4.1
Prove MDCE is a cyclic quadrilateral.
(3)
4.2
Prove that 𝑀𝐶 2 = 𝑀𝐵 2 + 𝐷𝐶 2 − 𝐷𝐵 2
(3)
4.3
Calculate CE if AB = 60 mm, ME = 40 mm and BC = 20 mm.
(4)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 38
QUESTION 5 (FEB/MARCH 2010)
5.1
Complete the statement:
The sum of the angles around a point is ...
5.2
(1)
In the figure below, O is the centre of the circle. K, L, M and N are points on the
circumference of the circle such that LM = MN. 𝐿𝑂̂𝑁 = 100°.
Determine, with reasons, the values of the following:
5.2.1
̂𝑁
𝐿𝑀
(3)
5.2.2
̂𝑀
𝐿𝐾
(3)
[7]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 39
QUESTION 6 (FEB/MARCH 2010)
6.1
Complete the following statement:
The angle between the tangent and the chord …
6.2
(1)
In the diagram below, two circles have a common tangent TAB. PT is a tangent to
the smaller circle. PAQ, QRT and NAR are straight lines. Let 𝑄̂ = 𝑥
6.2.1
Name, with reasons, THREE other angles equal to 𝑥.
(5)
6.2.2
Prove that APTR is a cyclic quadrilateral.
(5)
[11]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 40
QUESTION 7 (FEB/MARCH 2010)
Two circles touch each other at point A. The smaller circle passes through O, the centre of the
larger circle. Point E is on the circumference of the smaller circle. A, D, B and C are points on
the circumference of the larger circle.
OE || CA.
7.1
Prove, with reasons, that AE = BE.
(2)
7.2
Prove that ΔAED ||| ΔCEB.
(3)
7.3
Hence, or otherwise, show that AE 2 = DE. CE.
(3)
7.4
If AE. EB = EF. EC, show that E is the midpoint of DF.
(3)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 41
QUESTION 8 (FEB/MARCH 2010)
ΔABC is a right-angled triangle with 𝐵̂ = 90° . D is a point on AC such that BD ⊥ AC and E is a
point on AB such that DE ⊥ AB. E and D are joined.
AD : DC = 3 : 2.
AD = 15 cm.
8.1
Prove that ΔBDA ||| ΔCDB.
(3)
8.2
Calculate BD (Leave your answer in surd form)
(3)
8.3
Calculate AE (Leave your answer in surd form).
(6)
[12]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 42
QUESTION 9 (NOVEMBER 2011)
9.1
In the diagram below, O is the centre of the circle. PQ is a tangent to the circle at A.
B and C are points on the circumference of the circle. AB, AC and BC are joined.
̂ P = AB
̂C
Prove the theorem that states CA
(5)
9.2
RS is a diameter of the circle with centre O. Chord ST is produced to W. Chord SP
̂ 1 = 50°
produced meets the tangent RW at V. R
Calculate the size of:
9.2.1
̂S
WR
(1)
9.2.2
̂
W
(2)
9.2.3
𝑃̂1
(3)
9.2.4
̂1 = PT
̂S
Prove that V
(4)
[15]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 43
QUESTION 10 (NOVEMBER 2011)
AB is a diameter of the circle ABCD. OD is drawn parallel to BC and meets AC in E.
If the radius is 10 cm and AC = 16 cm, calculate the length of ED.
[5]
QUESTION 11 (NOVEMBER 2011)
CD is a tangent to circle ABDEF at D. Chord AB is produced to C. Chord BE cuts chord
̂4 = 𝑥 and 𝐷
̂1 = 𝑦.
AD in H and chord FD in G. AC || FD and FE = AB. Let 𝐷
11.1
Determine THREE other angles that are each equal to 𝑥.
(6)
11.2
Prove that ΔBHD ||| ΔFED.
(5)
11.3
Hence, or otherwise, prove that AB.BD = FD.BH.
(2)
[13]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 44
QUESTION 12 (NOVEMBER 2011)
ABCD is a parallelogram with diagonals intersecting at F. FE is drawn parallel to CD.
AC is produced to P such that PC = 2AC and AD is produced to Q such that DQ = 2AD.
12.1
Show that E is the midpoint of AD.
(2)
12.2
Prove PQ || FE.
(3)
12.3
If PQ is 60 cm, calculate the length of FE.
(5)
[10]
QUESTION 13 (FEB/MARCH 2011)
O is the centre of the circle. AB produced and DO produced meet at C.
BC = OA and AĈO = 22°
̂ D.
Calculate, with reasons, AO
BY MR M. SHABALALA
[5]
@NOMBUSO HIGH
Page 45
QUESTION 14 (FEB/MARCH 2011)
14.1
In the figure below O is the centre of the circle and PRST is a cyclic quadrilateral.
Prove the theorem that states
̂ S + PT
̂S = 180°
Prove the theorem that states PR
(5)
14.2
In the diagram below two circles intersect one another at D and B. AB is a straight
line such that it intersects the circle BCD at point E. BC is a straight line such that it
̂ 𝟐 = 𝟏𝟖𝟎° − 𝟐𝒙. FC = FD
intersects the circle ABD at F. DE, DB and DF are joined. 𝑭
14.2.1
14.2.2
Calculate, with reasons, in terms of 𝑥:
(a)
̂B
DE
(3)
(b)
𝐴̂
(2)
Hence, or otherwise, prove ED || BC.
(3)
[13]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 46
QUESTION 15 (FEB/MARCH 2011)
In the figure below DE || FG || BC.
AD = 36 cm, DF = 24 cm, AE = 48 cm and DE = GC = 40 cm.
Determine, with reasons, the lengths of:
15.1
EG
(2)
15.2
BC
(4)
[6]
QUESTION 16 (FEB/MARCH 2011)
In the diagram below DA is a tangent to the circle ACBT at A. CT and AD are produced to
meet at P. BT is produced to cut PA at D. AC, CB, AB and AT are joined. AC || BD
Let 𝐴̂1 = 𝑥
16.1
Prove that ΔABC ||| ΔADT.
(6)
16.2
Prove that PT is a tangent to the circle ADT at T.
(3)
16.3
Prove that ΔAPT ||| ΔTPD.
(3)
16.4
If AD =
2
3
𝐴𝑃 , show that 𝐴𝑃2 = 3𝑃𝑇 2
(4)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 47
QUESTION 17 (NOVEMBER 2012)
17.1
̂ = 𝐺̂ , prove the theorem that
If in Δ LMN and Δ FGH it is given that 𝐿̂ = 𝐹̂ and 𝑀
states
17.2
𝐿𝑀
𝐹𝐺
=
𝐿𝑁
𝐹𝐻
.
In the diagram below, ΔVRK has P on VR and T on VK such that PT || RK. VT = 4 units,
PR = 9 units, TK = 6 units and VP = 2𝑥 – 10 units.
Calculate the value of 𝑥.
(4)
[11]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 48
QUESTION 19 (NOVEMBER 2012)
O is the centre of the circle CAKB.
AK produced intersects circle AOBT at T.
𝐴𝐶̂ 𝐵 = 𝑥
19.1
Prove that 𝑇̂ = 180° − 2𝑥
(3)
19.2
Prove that AC || KB.
(5)
19.3
Prove ΔBKT ||| ΔCAT
(3)
19.4
If AK : KT = 5 : 2 , determine the value of
BY MR M. SHABALALA
AC
KB
@NOMBUSO HIGH
(3)
Page 49
[14]
QUESTION 20 (NOVEMBER 2012)
In the diagram below, O is the centre of the circle. Chord AB is perpendicular to diameter DC.
CM : MD = 4 : 9 and AB = 24 units.
20.1
Determine an expression for DC in terms of 𝑥 if CM = 4𝑥 units.
(1)
20.2
Determine an expression for OM in terms of 𝑥.
(2)
20.3
Hence, or otherwise, calculate the length of the radius.
(4)
[7]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 50
QUESTION 21 (FEB/MARCH 2013)
In the diagram below, O is the centre of the circle KTUV. PKR is a tangent to the circle at K.
̂ 𝑉 = 48° and K𝑇̂𝑈 = 120°
𝑂𝑈
Calculate, with reasons, the sizes of the following angles:
21.1
𝑉̂
(2)
21.2
𝐾𝑂̂𝑈
(2)
21.3
̂2
𝑈
(2)
21.4
̂1
𝐾
(2)
̂2
21.5 𝐾
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 51
QUESTION 22 (FEB/MARCH 2013)
22.1
Use the diagram below to prove the theorem which states that if VW || YZ.
Then
XV
VY
=
XW
WZ
(6)
22.2
In ΔPQR below, B lies on PR such that 2PB = BR. A lies on PQ such that
PA : PQ = 3 : 8. BC is drawn parallel to AR.
area of ΔPRA
22.2.1
Write down the value of
22.2.2
Calculate the value of the ratio
(2)
area of ΔQRA
support your answer.
BD
BQ
. Show all working to
(5)
[13]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 52
QUESTION 23 (FEB/MARCH 2013)
In the figure AGDE is a semicircle. AC is the tangent to the semicircle at A and EG produced
intersects AC at B. AD intersects BE in F. AG = GD. 𝐸̂1 = 𝑥.
23.1
Write down, with reasons, FOUR other angles each equal to 𝑥.
(8)
23.2
Prove that BE.DE = AE.FE
(7)
23.3
̂1
Prove that 𝐵̂1 = 𝐷
(4)
[19]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 53
QUESTION 24 (NOVEMBER 2013)
In ΔADC, E is a point on AD and B is a point on AC such that EB || DC. F is a point on AD
such that FB || EC. It is also given that AB = 2BC.
24.1
Determine the value of AF : FE
(2)
24.2
Calculate the length of ED if AF = 8 cm.
(4)
[6]
QUESTION 25 (NOVEMBER 2013)
In the diagram below, O is the centre of the circle and OB is perpendicular to chord AC.
Prove, using Euclidean geometry methods, the theorem that states AB = BC.
[5]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 54
QUESTION 26 (NOVEMBER 2013)
In the diagram below, O is the centre of the circle. BD is a diameter of the circle. GEH is a
tangent to the circle at E. F and C are two points on the circle and FB, FE, BC, CE and BE are
drawn.
𝐸̂1 = 32° and 𝐸̂3 = 56°
Calculate, with reasons, the values of :
26.1
𝐸̂2
(2)
26.2
𝐸𝐵̂ 𝐶
(3)
26.3
𝐹̂
(4)
[9]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 55
QUESTION 27 (NOVEMBER 2013)
In the diagram below, two circles intersect at K and Y. The larger circle passes through O, the
centre of the smaller circle. T is a point on the smaller circle such that KT is a tangent to the
larger circle. TY produced meets the larger circle at W.
WO produced meets KT at E.
̂1 = 𝑥
Let 𝑊
27.1
Determine FOUR other angles, each equal to 𝑥.
(8)
27.2
Prove that 𝑇̂ = 90° − 𝑥.
(3)
27.3
Prove that KE = ET.
(3)
27.4
Prove that 𝐾𝐸 2 = 𝑂𝐸. 𝑊𝐸
(6)
[20]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 56
QUESTION 28 (FEB/MARCH 2014)
In the diagram, O is the centre of the circle. PWSR is a cyclic quadrilateral. PS, WO and OS are
drawn. PW || OS and 𝑃̂1 = 36°
Calculate the sizes of the following angles:
28.1
𝑆𝑂̂𝑊
(2)
28.2
̂2
𝑊
(2)
28.3
𝑂𝑆̂𝑊
(3)
28.4
𝑅̂
(3)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 57
QUESTION 29 (FEB/MARCH 2014)
In the diagram, OP is the diameter of the smaller circle. O is the centre of the larger circle.
Also, PTW is a chord of the larger circle and T lies on the smaller circle. OT is joined.
If OT = 10 cm and PW = 48 cm, calculate the length of the radius of the smaller circle.
[5]
QUESTION 30 (FEB/MARCH 2014)
In the diagram, PSW and WT are tangents to circle RST at S and T respectively.
PT is drawn and intersects the circle at R. RS and ST are joined. RT = TS.
Let 𝑆̂1 = 𝑥 and 𝑇̂2 = 𝑦.
30.1
Name, with reasons, THREE angles each equal to 𝑦.
(6)
30.2
Prove that ΔPRS ||| ΔPST.
(3)
30.3
Prove that PS × RT = RS × PT.
(3)
[12]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 58
QUESTION 31 (FEB/MARCH 2014)
31.1
In ΔABC below, D is a point on AB and E is a point on AC such that DE || BC. Prove,
using Euclidean geometry methods, the theorem that states
AD
DB
=
AE
EC
(7)
31.2
In the diagram below, ACH is a triangle with point B on AC and point G on AH
such that BG || CH. F is a point on AH and D is a point on HC such that
FD || AC. GB intersects FD at E.
It is also given that HD : DC = 5 : 3 and AB = 2BC.
If AH = 48 cm, calculate the following with reasons:
31.2.1
HF
(3)
31.2.2
FG
(3)
31.2.3
EF : ED
(2)
[15]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 59
QUESTION 32 (NOVEMBER 2014)
32.1
In the diagram, O is the centre of the circle passing through A, B and C.
𝐶𝐴̂𝐵 = 48° , 𝐶𝑂̂𝐵 = 𝑥 and 𝐶̂2 = 𝑦
Determine, with reasons, the size of:
32.2
32.1.1
𝑥
(2)
32.1.2
𝑦
(2)
In the diagram, O is the centre of the circle passing through A, B, C and D. AOD
̂ 𝐹 = 30° and OF are joined
is a straight line and F is the midpoint of chord CD. 𝑂𝐷
Determine, with reasons, the size of:
32.2.1
𝐹̂1
(2)
32.2.2
𝐴𝐵̂ 𝐶
(2)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 60
32.3
In the diagram, AB and AE are tangents to the circle at B and E respectively.
BC is a diameter of the circle. AC = 13, AE= 𝑥 and BC = 𝑥 + 7
32.3.1
Give reasons for the statements below.
(2)
32.3.2
Calculate the length of AB.
(4)
[14]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 61
QUESTION 33 (NOVEMBER 2014)
33.1
In the diagram, points D and E lie on sides AB and AC of ΔABC respectively such
that DE || BC. DC and BE are joiced
33.1.1
Explain why the areas of ΔDEB and ΔDEC are equal.
33.1.2
Given below is the partially completed proof of the theorem that
states if in any ΔABC the line DE || BC then
AD
DB
=
(1)
AE
EC
Using the above diagram, complete the proof of the theorem
Construction: Construct altitudes (heights) ℎ and 𝑘 in ΔADE
(5)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 62
33.2
In the diagram, ABCD is a parallelogram. The diagonals of ABCD intersect in M.
F is a point on AD such that AF : FD = 4 : 3. E is a point on AM such that EF || BD.
FC and MD intersect in G.
Calculate, giving reasons, the ratio of :
33.2.1
33.2.2
33.2.3
EM
(3)
AM
CM
(3)
ME
area ΔFDC
(4)
area ΔBDC
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 63
QUESTION 34 (NOVEMBER 2014)
The two circles in the diagram have a common tangent XRY at R. W is any point on the small
circle. The straight line RWS meets the large circle at S. The chord STQ is a tangent to the small
̂4 = 𝑥 and 𝑅̂2 = 𝑦
circle, where T is the point of contact. Chord RTP is drawn. Let 𝑅
34.1
Give reasons for the statements below.
̂4 = 𝑥 and 𝑅̂2 = 𝑦
Let 𝑅
statement
34.1.1
𝑇̂3 = 𝑥
34.1.2
𝑃̂1 = 𝑥
34.1.3
WT∥SP
34.1.4
𝑆̂1 = 𝑦
34.1.5
𝑇̂2 = 𝑦
Reason
(5)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 64
WR.RP
34.2
Prove that RT =
34.3
Identify, with reasons, another TWO angles equal to 𝑦.
(4)
34.4
̂3 = 𝑊
̂2
Prove that 𝑄
(3)
34.5
Prove that ΔRTS ||| ΔRQP.
(3)
34.6
Hence, prove that
(2)
RS
𝑊𝑅
𝑅𝑄
𝑅𝑆 2
= 𝑅𝑃2
(3)
[20]
QUESTION 35 (FEB/MARCH 2015)
In the diagram, AB is a chord of the circle with centre O. M is the midpoint of AB.
MO is produced to P, where P is a point on the circle. OM= 𝑥 units, AB= 20 units
And
PM
OM
=
5
2
35.1
Write down the length of MB.
(1)
35.2
Give a reason why OM ⊥ AB.
(1)
35.3
Show that OP =
35.4
Calculate the value of 𝑥.
3𝑥
2
units.
(2)
(3)
[7]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 65
QUESTION 36 (FEB/MARCH 2015)
In the diagram below, the circle with centre O passes through A, B, C and D. AB ∥ DC and
̂ C = 110°. The chords AC and BD intersect at E. EO, BO, CO, and BC are joined.
BO
36.1
36.2
Calculate the size of the following angles, giving reasons for your answers:
36.1.1
̂
𝐷
(2)
36.1.2
𝐴̂
(2)
36.1.3
𝐸̂2
(4)
Prove that BEOC is a cyclic quadrilateral.
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 66
QUESTION 37 (FEB/MARCH 2015)
37.1
In the figure, TRSW is a cyclic quadrilateral with TW = WS. RT and RS are produced to
meet tangent VWZ at V and Z respectively. PRQ is a tangent to the circle at R. RW is
joined. 𝑅̂2 = 30° and 𝑅̂4 = 50°
37.3.1
̂3 = 30°.
Give a reason why 𝑅
(1)
37.3.2
State, with reasons, TWO other angles equal to 30°
(3)
37.3.3
Determine, with reasons, the size of:
37.3.4
(a)
𝑆̂2
(3)
(b)
𝑉̂
(4)
Prove that WR2 = RV × RS.
(5)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 67
QUESTION 38 (FEB/MARCH 2015)
̂ = 90°. NP is drawn parallel to TR with N on TM and P on RM. It is further
In ΔTRM, 𝑀
given that RT = 3PN
38.1
Give reasons for the statements below
Statement
Reason
In ΔPNM and ΔRTM:
……………………………………………….
̂1 = 𝑇̂
38.1.1 𝑁
̂ is common
𝑀
38.1.2 ∴ ΔPNM ||| ΔRTM
……………………………………………….
(2)
PM
=
1
38.2
Prove that
38.3
Show that RN 2 − PN 2 = 2RP 2 .
RM
(2)
3
(4)
[8]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 68
QUESTION 39 (JUNE 2015)
In the diagram below, PR is a chord of the circle with centre O. Diameter ST is perpendicular to
PR at M. PR= 8 cm ,
MT = 2 cm, OM = 𝑥 cm.
39.1
Write OP in terms of 𝑥 and a number
(1)
39.2
Write down the length of PM. Give a reason.
(2)
39.3
Hence calculate the length of the radius of the circle.
(3)
[6]
QUESTION 40 (JUNE 2015)
In the diagram below, the vertices of ΔPNR lie on the circle with centre O. Diameter SR
is perpendicular to chord NP at T.
Point W lies on NR. 𝑅̂2 = 30° .
40.1
40.2
Calculate the size of the following angles, giving reasons for your answers:
40.1.1
𝑆̂
(3)
40.1.2
𝑅̂1
(3)
40.1.3
̂1
𝑁
(3)
If it is further given that NW= WR , prove that TNWO is a cyclic quadrilateral
(4)
[13]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 69
QUESTION 41 (JUNE 2015)
In the diagram below, P, A, Q, R and S lie on the circle with centre O. SB is a tangent to the
circles at S and RW= WP. AOWS and RWP are straight lines.
41.1
Write down, with reasons, the size of the following angles:
41.1.1
𝐵𝑆̂𝑊
(2)
41.1.2
̂1
𝑊
(2)
41.2
Why is SB ∥ RP ?
(1)
41.3
Prove, with reasons, that:
41.3.1
ΔAPS ||| ΔRWS
(4)
41.3.2
RS 2 = WS. AS
(4)
41.3.3
𝐴𝑆 =
𝑅𝑊 2
𝑊𝑆
+ 𝑊𝑆
(3)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 70
QUESTION 42 (SEPTEMBER 2015)
In the diagram below, PQT is a tangent to the larger circle ABQ at Q. A smaller circle intersects
the larger circle at A and Q. BAP and BQR are straight lines with P and R on the smaller circle.
AQ and PR are drawn.
42.1
Prove that PQ = PR
(7)
42.2
Prove that ΔPBQ ||| ΔPQA
(4)
42.3
Prove that the lengths of PA, PR and PB(in this order) form a geometric sequence.(3)
[14]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 71
QUESTION 43 (NOVEMBER 2015)
In the diagram below, cyclic quadrilateral ABCD is drawn in the circle with centre O.
43.1.1 Complete the following statement:
The angle subtended by a chord at the centre of a circle is … the angle subtended
by the same chord at the circumference of the circle.
(1)
43.1.2 Use QUESTION 43.1.1 to prove that 𝐴̂ + 𝐶̂ = 180°
(3)
43.2
In the diagram below, CD is a common chord of the two circles. Straight lines
ADE and BCF are drawn. Chords AB and EF are drawn.
Prove that EF ∥ AB
(5)
[9]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 72
QUESTION 44 (NOVEMBER 2015)
In the diagram below, ΔABC is drawn in the circle. TA and TB are tangents to the circle. The
straight line THK is parallel to AC with H on BA and K on BC. AK is drawn.
Let 𝐴̂3 = 𝑥
44.1
̂3 = 𝑥
Prove that 𝐾
(4)
44.2
Prove that AKBT is a cyclic quadrilateral.
(2)
44.3
̂ 𝐵.
Prove that TK bisects 𝐴𝐾
(4)
44.4
Prove that TA is tangent to the circle passing through the points A, K and H.
(2)
44.5
S is a point in the circle such that the points A, S, K and B are concyclic.
Explain why A, S, B and T are also concyclic.
(2)
[14]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 73
QUESTION 45 (NOVEMBER 2015)
In the diagram below, BC = 17 units, where BC is a diameter of the circle. The length of chord
BD is 8 units. The tangent at B meets CD produced at A.
45.1
Calculate, with reasons, the length of DC.
(3)
45.2
E is a point on BC such that BE : EC = 3 : 1. EF is parallel to BD with F on DC
45.2.1
Calculate, with reasons, the length of CF.
(3)
45.2.2
Prove that ΔBAC ||| ΔFEC.
(5)
45.2.3
Calculate the length of AC.
(4)
45.2.4
Write down, giving reasons, the radius of the circle passing
through points A, B and C.
(2)
[17]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 74
QUESTION 46 (NOVEMBER 2015)
46.1
Complete the following statement:
If the sides of two triangles are in the same proportion, then the triangles are…
46.2
(1)
In the diagram below, K, M and N respectively are points on sides PQ, PR and
QR of ΔPQR. KP = 1,5; PM= 2; KM= 2,5; MN= 1; MR=1,25 and
NR = 0,75.
46.2.1
Prove that ΔKPM ||| ΔRNM.
(3)
46.2.2
Determine the length of NQ.
(6)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 75
QUESTION 47 (FEB/MARCH 2016)
47.1
In the diagram below, tangent KT to the circle at K is parallel to the chord NM.
̂2 = 40° and 𝑀𝐾
̂ 𝑇 = 84°.
NT cuts the circle at L. ΔKML is drawn. 𝑀
Determine, giving reasons, the size of :
47.1.1
̂2
𝐾
(2)
47.1.2
̂1
𝑁
(3)
47.1.3
𝑇̂
(2)
47.1.4
𝐿̂2
(2)
47.1.5
𝐿̂1
(1)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 76
47.2
In the diagram below, AB and DC are chords of a circle. E is a point on AB such that
BCDE is a parallelogram. 𝐷𝐸̂ 𝐵 = 108° and 𝐷𝐴̂𝐸 = 2𝑥 + 40°.
Calculate , giving reasons, the value of 𝑥.
(5)
[15]
QUESTION 48 (FEB/MARCH 2016)
In the diagram below, EO bisects side AC of ΔACE. EDO is produced to B such that
BO = OD. AD and CD produced meet EC and EA at G and F respectively.
48.1
Give a reason why ABCD is a parallelogram.
(1)
48.2
Write down, with reasons, TWO ratios each equal to
48.3
Prove that 𝐴̂1 = 𝐹̂2 .
48.4
It is further given that ABCD is a rhombus. Prove that ACGF is a cyclic
𝐸𝐷
𝐷𝐵
(4)
(5)
quadrilateral
(3)
[13]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 77
QUESTION 49 (FEB/MARCH 2016)
49.1
In the diagram below, ΔABC and ΔPQR are given with 𝐴̂ = 𝑃̂ , 𝐵̂ = 𝑄̂ and 𝐶̂ = 𝑅̂
DE is drawn such that AD = PQ and AE = PR.
49.1.1
Prove that ΔADE ≡ ΔPQR
(2).
49.1.2
Prove that DE ∥ BC.
(3)
49.1.3
Hence, prove that
BY MR M. SHABALALA
𝐴𝐵
𝑃𝑄
=
𝐴𝐶
𝑃𝑅
@NOMBUSO HIGH
(2)
Page 78
49.2
In the diagram below, VR is a diameter of a circle with centre O. S is any point on the
circumference. P is the midpoint of RS. The circle with RS as diameter cuts VR at T.
ST, OP and SV are drawn.
49.2.1
Why is OP ⊥ PS?
(1)
49.2.2
Prove that ΔROP ||| ΔRVS.
(4)
49.2.3
Prove that ΔRVS ||| ΔRST.
(3)
49.2.4
Prove that ST 2 = VT. TR
(6)
[21]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 79
QUESTION 50 (SEPTEMBER 2016)
50.1
In the diagram below, BAED is a cyclic quadrilateral with BA || DE. BE = DE and
𝐴𝐸̂ 𝐷 = 70°. The tangent to the circle at D meets AB produced at C
Calculate, with reasons the sizes of the following.
50.1.1
𝐴̂
(2)
50.1.2
𝐵̂1
(2)
50.1.3
̂2
𝐷
(2)
50.1.4
𝐵̂2
(2)
50.1.5
̂1
𝐷
(3)
[11]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 80
QUESTION 51 (SEPTEMBER 2016)
In the diagram below, SP is a tangent to the circle at P and PQ is a chord. Chord QF produced
meets SP at S and chord RP bisects 𝑄𝑃̂𝑆 . PR produced meets QS at B. BC || SP and cuts the
chord QR at D. QR produced meets SP at A. Let 𝐵̂2 = 𝑥
51.1
Name, with reasons, 3 angles equal to 𝑥.
(4)
51.2
Prove that PC = BC
(2)
51.3
Prove that RCQB is a cyclic quadrilateral.
(2)
51.4
Prove that ΔPBS ||| ΔQCR.
(5)
51.5
Show that PB.CR = QB.CP
(4)
[17]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 81
QUESTION 52 (NOVEMBER 2016)
52.1
52.2
In the diagram below PQRT is a cyclic quadrilateral having RT ∥ QP. The tangent at P
meets RT produced at S. QP = QT and 𝑃𝑇̂𝑆 = 70°
52.1.1
Give a reason why 𝑃̂2 = 70°
52.1.2
Calculate, with reasons, the size of :
(1)
(a)
𝑄̂1
(3)
(b)
𝑃̂1
(2)
A, B and C are points on the circle having centre O. S and T are points on AC
and AB respectively such that OS ⊥ AC and OT ⊥ AB. AB= 40 and AC = 48
52.2.1
Calculate AT.
52.2.2
If OS =
7
15
(1)
OT, calculate the radius OA of the circle.
(5)
[12]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 82
QUESTION 53 (NOVEMBER 2016)
ABC is a tangent to the circle BFE at B. From C a straight line is drawn parallel to BF to meet
FE produced at D. EC and BD are drawn. 𝐸̂1 = 𝐸̂2 = 𝑥 and 𝐶̂2 = 𝑦
53.1
Give a reason why EACH of the following is true:
53.1.1
𝐵̂1 = 𝑥
(1)
53.1.2
𝐵𝐶̂ 𝐷 = 𝐵̂1
(1)
53.2
Prove that BCDE is a cyclic quadrilateral
(2)
53.3
Which TWO other angles are each equal to 𝑥?
(2)
53.4
Prove that 𝐵̂2 = 𝐶̂1
(3)
[9]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 83
QUESTION 54 (NOVEMBER 2016)
54.1
In the diagram ΔPQR is drawn. S and T are points on sides PQ and PR respectively such
that ST ∥ QR.
Prove the theorem which states that
BY MR M. SHABALALA
PS
SQ
=
PT
TR
@NOMBUSO HIGH
(6)
Page 84
54.2
In the diagram HLKF is a cyclic quadrilateral. The chords HL and FK are produced to
meet at M. The line through F parallel to KL meets MH produced at G.
MK = 𝑥, KF = 2𝑥, ML = 𝑦 and LH = HG.
54.2.1
̂𝑀
Give a reason why 𝐺𝐹̂ 𝑀 = 𝐿𝐾
54.2.2
Prove that :
(a)
GH = 𝑦
(3)
(b)
ΔMFH ||| ΔMGF
(5)
(c)
54.2.3
(1)
𝐺𝐹
𝐹𝐻
Show that
=
𝑦
𝑥
3𝑥
(2)
2𝑦
=√
3
2
(3)
[20]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 85
QUESTION 55 (FEB/MARCH 2017)
In the diagram, PQRS is a cyclic quadrilateral. ST is a tangent to the circle S and chord SR is
produced to V. PQ = QR, 𝑆̂1 = 42° and 𝑆̂2 = 108°
Determine, with reasons, the size of the following angles:
55.1
𝑄̂
(2)
55.2
𝑅̂2
(2)
55.3
𝑃̂2
(2)
55.4
𝑅̂3
(2)
[8]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 86
QUESTION 56 (FEB/MARCH 2017)
In the diagram, PQRS is a quadrilateral with diagonals PR and QS drawn. W is a point on PS.
WT is parallel to PQ with T on QS. WV is parallel to PR with V on RS. TV is drawn.
PW : WS = 3 : 2.
56.1
Write down the value of the following ratios:
56.1.1
56.1.2
ST
(2)
TQ
SV
(1)
VR
56.2
Prove that 𝑇̂1 = 𝑄̂1
(4)
56.3
Complete the following statement: ΔVWS ||| Δ …
(1)
56.4
Determine WV : PR
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 87
QUESTION 57 (FEB/MARCH 2017)
57.1
Proof for angle at the centre = 2x angle at the circumference ……………it was
[ 5 marks]
57.2
In the diagram, O is the centre of the circle and P, Q, S and R are points on the circle.
PQ = QS and 𝑄𝑅̂ 𝑆 = 𝑦. The tangent at P meets SQ produced at T. OQ intersects
PS at A.
57.2.1
Give a reason why 𝑃̂2 = 𝑦
(1)
57.2.2
Prove that PQ bisects 𝑇𝑃̂𝑆.
(4)
57.2.3
Determine 𝑃𝑂̂ 𝑄 in terms of 𝑦.
(2)
57.2.4
Prove that PT is a tangent to the circle that passes through points
57.2.5
P, O and A.
(2)
Prove that 𝑂𝐴̂𝑃 = 90°.
(5)
[19]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 88
QUESTION 58 (FEB/MARCH 2017)
In the diagram, LK is a diameter of the circle with centre P. RNS is a tangent to the circle at N. T
T is a point on NK and TP ⊥ KL. 𝑃𝐿̂𝑁 = 𝑥.
58.1
Prove that TPLN is a cyclic quadrilateral.
(3)
58.2
̂1 in terms of 𝑥.
Determine, giving reasons, the size of 𝑁
(3)
58.3
Prove that:
58.3.2
ΔKTP ||| ΔKLN
(3)
58.3.2
KT. KN = 2KT 2 − 2TP 2
(5)
[14]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 89
QUESTION 59 (NOVEMBER 2017)
In the diagram, points A, B, D and C lie on a circle. CE ∥ AB with E on AD produced.
̂2 = 50° and 𝐶̂1 = 15°
Chords CB and AD intersect at F. 𝐷
59.1
59.2
Calculate, with reasons, the size of:
59.1.1
𝐴̂
(3)
59.1.2
𝐶̂2
(2)
Prove, with a reason, that CF is a tangent to the circle passing through points
C, D and E.
(2)
[7]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 90
QUESTION 60 (NOVEMBER 2017)
In the diagram, ΔABC and ΔACD are drawn. F and G are points on sides AB and AC
respectively such that AF = 3𝑥, FB = 2𝑥, AG = 12𝑦 and GC = 8𝑦. H, E and K are points on
side AD such that GH ∥ CK and GE ∥ CD.
60.1
Prove that :
60.1.1
60.1.2
60.2
FG ∥ BC
AH
HK
=
(2)
AE
(3)
ED
It is further given that AH = 15 and ED = 12, calculate the length of EK.
(5)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 91
QUESTION 61 (NOVEMBER 2017)
In the diagram, W is a point on the circle with centre O. V is a point on OW. Chord MN is drawn
such that MV = VN. The tangent at W meets OM produced at T and ON produced at S.
61.1
Give a reason why OV ⊥ MN
61.2
Prove that:
(1)
61.2.1
MN ∥ TS
(2)
61.2.2
TMNS is a cyclic quadrilateral
(4)
61.2.3
OS.MN = 2ON.WS
(5)
[12]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 92
QUESTION 62 (NOVEMBER 2017)
62.1
Proof for tan chord theorem….
62.2
In the diagram, BC is a diameter of the circle. The tangent at point D on the circle
[5 marks]
meets CB produced at A. CD is produced to E such that EA ⊥ AC
Let 𝐶̂ = 𝑥.
62.2.1
62.2.2
62.2.3
Give a reason why:
(a)
̂3 = 90°
𝐷
(1)
(b)
ABDE is a cyclic quadrilateral
(1)
(c)
̂2 = 𝑥
𝐷
(1)
Prove that:
(a)
AD = AE
(3)
(b)
ΔADB ||| ΔACD
(3)
It is further given that BC = 2𝐴𝐵 = 2𝑟
(a)
Prove that 𝐴𝐷 2 = 3𝑟 2
(2)
(b)
Hence, prove that ΔADE is equilateral.
(4)
[20]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 93
QUESTION 63 (FEB/MARCH 2018)
In the diagram, PQRT is a cyclic quadrilateral in a circle such that PT = TR. PT and QR
produced to meet in S. TQ is drawn. 𝑆𝑄̂ 𝑃 = 70°
63.1
63.2
Calculate, with reasons, the size of :
63.1.1
𝑇̂1
(2)
63.1.2
𝑄̂1
(2)
If it is further given that PQ ∥ TR:
63.2.1
Calculate, with reasons, the size of 𝑇̂2
63.2.2
Prove that
TR
TS
=
RQ
RS
(2)
(2)
[8]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 94
QUESTION 64 (FEB/MARCH 2018)
In the diagram, PR is a diameter of the circle with centre O. ST is a tangent to the circle at T and
meets RP produced at S. 𝑆𝑃̂ 𝑇 = 𝑥 and 𝑆̂ = 𝑦
Determine, with reasons, 𝑦 in terms of 𝑥
[6]
QUESTION 65(GP)
ALB is a tangent to circle LMNP. ALB||MP.
Prove that:
65.1
LM = LP
(4)
65.2
̂P
LN bisects MN
(4)
65.3
LM is a tangent to circle MNQ
(4)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 95
QUESTION 66 (FEB/MARCH 2018)
In the diagram, DEFG is a quadrilateral with DE = 45 and GF = 80. The diagonals GE and DF
̂ 𝐸 = 𝐹𝐸̂ 𝐺 and 𝐷𝐺̂ 𝐸 = 𝐸𝐹̂ 𝐺
meet in H. 𝐺𝐷
66.1
Give a reason why ΔDEG ||| ΔEGF
(1)
66.2
Calculate the length of GE.
(3)
66.3
Prove that ΔDEH ||| ΔFGH.
(3)
66.4
Hence, calculate the length of GH
(3)
[10]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 96
QUESTION 67 (FEB/MARCH 2018)
67.1
Proof angle at the centre = 2 × angle at circumference.
67.2
In the diagram, the circle with centre F is drawn. Points A, B, C and D lie on the
[5 marks]
circle. Chords AC and BD intersect at E such that EC = ED. K is the midpoint of
chord BD. FK, AB, CD, AF, FE and FD are drawn. Let 𝐵̂ = 𝑥.
67.2.1
Determine, with reasons, the size of EACH of the following in
terms of 𝑥:
(a)
𝐹̂1
(2)
(b)
𝐶̂
(2)
67.2.2
Prove, with reasons, that AFED is a cyclic quadrilateral.
(4)
67.2.3
Prove, with reasons, that 𝐹̂3 = 𝑥
(6)
67.2.4
If area ΔAEB = 6, 25 × area ΔDEC, calculate
AE
ED
(5)
[24]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 97
QUESTION 68 (SEPTEMBER 2018)
In the diagram below, AOCD is a diameter of the circle with centre O and chord BE = 30 cm.
AOCD ⊥ BE and OC = 2CD.
Calculate with reasons:
68.1
BC
(2)
68.2
If CD = 𝑎 ubits, determine OC in terms of 𝑎.
(1)
68.3
Calculate OB
(1)
68.4
AB (correct to one decimal place)
(3)
68.5
The radius of the circle CAB.
(2)
[9]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 98
QUESTION 69 (SEPTEMBER 2018)
69.1
In the diagram below, ABCD is a cyclic quadrilateral of the circle with centre O.
̂ = 180° .
Use the diagram to prove the theorem which states that 𝐵̂ + 𝐷
(5)
69.2
KOL is the diameter of the circle KPNML having centre O. R is the point on chord KN,
̂ 𝑁 and cuts LP in T.
such that KR = RO. OR is produced to P. Chord KM bisects 𝐿𝐾
̂1 = 𝑥
𝐾
Prove with reasons that:
69.2.1
69.2.2
69.2.3
TK = TL
KOTP is a cyclic quadrilateral.
PN ∥ MK
(5)
(3)
(3)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 99
QUESTION 70 (SEPTEMBER 2018)
In ΔABC, R is a point on AB. S and P are points on AC such that RS // BP. P is the midpoint of
AC. RC and BP intersects at T.
AR
AB
=
3
5
Calculate with reasons, the following ratios:
70.1
70.2
70.3
AS
(3)
SC
RT
(2)
TC
∆ARS
(3)
∆ABC
[8]
QUESTION 71 (SEPTEMBER 2018)
In the diagram alongside,
ACBT is a cyclic quadrilateral.
BT is produced to meet tangent AP on D.
CT is produced to P. AC // DB.
71.1
Prove that 𝑃𝐴2 = 𝑃𝑇. 𝑃𝐶
71.2
If PA = 6 units, TC = 5 units
(5)
and PT = 𝑥, show that
𝑥 2 + 5𝑥 − 36 = 0
(2)
71.3
Calculate the length of PT.
(2)
71.4
Calculate the length of PD.
(3)
[12]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 100
QUESTION 72 (NOVEMBER 2018)
72.1
PON is a diameter of the circle at O. TM is a tangent to the circle at M, a point on the
circle. R is another point on the circle such that OR ∥ PM. NR and MN are drawn.
̂1 = 66°.
Let 𝑀
Calculate, with reasons, the size of EACH of the following angles:
72.1.1
𝑃̂
(2)
72.1.2
̂2
𝑀
(2)
72.1.3
̂1
𝑁
(1)
72.1.4
𝑂̂2
(2)
72.1.5
̂2
𝑁
(3)
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 101
72.2
In the diagram, ∆AGH is drawn. F and C are points on AG and AH respectively such that
AF = 20 units, FG = 15 units and CH = 21 units. D is a point on FC such that ABCD
is a rectangle with AB also parallel to GH. The diagonals of ABCD intersect at M, a
point on AH.
72.2.1
Explain why FC∥GH.
(1)
72.2.2
Calculate, with reasons, the length of DM.
(5)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 102
QUESTION 73 (NOVEMBER 2018)
73.1
Proof for opposite angles in a cyclic quadrilateral
(5 marks)
73.2
In the diagram, a smaller circle ABTS and a bigger circle BDRT are given. BT is
a common chord. Straight lines STD and ATR are drawn. Chords AS and DR are
produced to meet in C, a point outside the two circles. BS and BD are drawn.
𝐴̂ = 𝑥 and 𝑅̂1 = 𝑦.
73.2.1
Name, giving a reason, another angle equal to:
(a)
𝑥
(2)
(b)
𝑦
(2)
73.2.2
Prove that SCDB is a cyclic quadrilateral.
73.2.3
̂2 = 30°
It is further given that 𝐷
(3)
and 𝐴𝑆̂𝑇 = 100°.
Prove that SD is not a diameter of circle BDS.
(4)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 103
QUESTION 74 (NOVEMBER 2018)
In the diagram, ABCD is a cyclic quadrilateral such that AC ⊥ CB and DC = CB. AD is
produced to M such that AM ⊥ MC. Let 𝐵̂ = 𝑥.
74.1
74.2
Prove that:
74.1.1
MC is a tangent to the circle at C
(5)
74.1.2
ΔACB ||| ΔCMD.
(3)
Hence, or otherwise, prove that:
74.2.1
74.2.2
CM2
DC2
𝐴𝑀
𝐴𝐵
=
AM
(6)
AB
= 𝑠𝑖𝑛2 𝑥
(2)
[16]
BY MR M. SHABALALA
@NOMBUSO HIGH
Page 104