f
EUCLIDEAN
GEOMETRY
GRADE 12
PAST EXAM PAPERS
CELLPHONE NUMBER: 0733318802 Page 1
EMAILBY
ADDRESS:
melulekishabalala@gmail.com
MR M. SHABALALA
@NOMBUSO
GR 12 TWO NEW PROOFS TO BE KNOWN FOR EXAM
PROOF - PROPORTIONALITY THEOREM
7 Marks
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
t
tt
CONSTRUCTION : Draw altitude h
㌠
to base AD and altitude k
AE. Join DC and BE
th
PROOF:
th
tt
th
t
th
㌠ t
But th
th
th
tt
Similarly
tt
㌠
㌠
ಸཪ
BY MR M. SHABALALA
t
tt
tt
㌠
㌠
tt
t
tt
t
t
t
t
th
th
t
th
㌠ t
t
㌠
tt
㌠
to base

(same height h)
S 
(same height k)
S/R
㌠ t (Same base between DE//BC) S 
S
(7)
@NOMBUSO
Page 2
PROOF - SIMILARITY THEOREM
7 Marks
2. If two triangles are equiangular(AAA), then the corresponding sides are in
Proportion.
Reason (||| Δs
OR
equiangular Δs)
Given
Prove
: ΔABC and ΔPQR with A
:
t
㌠
P , B
t㌠
Q and C
R
Construction: X and Y on AB, AC respectively such that AX = PQ
 all 3 sides
and AY = PR. Join XY.
Proof
: In ΔAXY and ΔPQR
AX = PQ construction
AY PR
construction
A P
given
th
AXY
AXY
R
R
(SAS)
Reason
(
S
(
congruent)
t
given)
S
XY BC
( corresponding
AB
(Prop Int theorem XY || BC) S/R
AX
t
AC
AY
㌠
)
AX = PQ and AY = PR
Reason
S
(7)
Similarly it can be proved that ΔABC ||| ΔPQR
t
㌠
BY MR M. SHABALALA
t㌠
@NOMBUSO
Page 3
QUESTION 1
1 complete the following statement
1.1 The exterior angle of a cyclic quadrilateral is equals to…
…………………………………………………………….
(1)
1.2 Line from the centre of circle perpendicular to the
chord … …………………………………………………………………..
(1)
1.3 Line from the centre of circle to the midpoint of the
chord … ………………………………………………………………. ..
(1)
1.4 The angle in semi-circle is ……………………….. ……. ..
(1)
1.5 The opposite angles of a cyclic quadrilateral
are …………………………………………………………………..
(1)
1.6 The angle between the tangent and a chord
is ………………………………………………………………… .
(1)
1.7 The angle subtended by an arc at the centre of the circle
is… ……………………………………………………………………
(1)
[7]
BY MR M. SHABALALA
@NOMBUSO
Page 4
PROOF 3 (LINE FROM THE CENTRE)
O is the centre of the circle and AB is a chord. OB
theorem which states that AB = BC
(99(
AC. Prove the
(5)
GIVEN
R.T.P
CONSTRUCTION
PROOF:
BY MR M. SHABALALA
@NOMBUSO
Page 5
PROOF 4(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.
t ㌠
(5)
GIVEN
R.T.P
CONSTRUCTION
PROOF:
BY MR M. SHABALALA
@NOMBUSO
Page 6
PROOF 5(CYCLIC QUAD PROOF)
In the diagram below, O is the centre of the circle. Use the diagram to prove the
h (5)
theorem which states that: If PQRS is a cyclic quadrilateral then
GIVEN: _______________________________
R.T.P
:_______________________________
CONSRUCTION: _______________________
PROOF:
PROOF 6( TANGENT PROOF)
In the diagram below the circle with centre O passes through points B, A,
and C. AE is a tangent to the circle at A. BC, AB and AC are joined.
Prove that t㌠= ㌠
(6)
GIVEN: _______________________________
R.T.P
:_______________________________
CONSRUCTION: _______________________
PROOF:
BY MR M. SHABALALA
@NOMBUSO
Page 7
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
If ㌠
, calculate, with reasons, the size of :
1.1
1.2
1.3
BD.
(3)
(2)
t
Show that AE bisects t t
BY MR M. SHABALALA
(3)
[8]
@NOMBUSO
Page 8
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
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.
2.2.1
‴.
(4)
2.2.2
Write down, with reasons, THREE other angles equal to ‴.
(3)
2.2.3
Determine, with reasons, ㌠t in terms of ‴.
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
Page 9
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
If DE = ‴ units and AB =
AE : EC = 4 : 7.
units, calculate ‴ 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
(3)
4.3
Calculate CE if AB = 60 mm, ME = 40 mm and BC = 20 mm.
㌠
t
t㌠
tt
(4)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 10
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.
.
Determine, with reasons, the values of the following:
5.2.1
(3)
5.2.2
(3)
[7]
BY MR M. SHABALALA
@NOMBUSO
Page 11
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
6.2.1
6.2.2
‴
Name, with reasons, THREE other angles equal to ‴.
Prove that APTR is a cyclic quadrilateral.
(5)
(5)
[11]
BY MR M. SHABALALA
@NOMBUSO
Page 12
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
7.4
If AE EB
DE CE.
EF EC, show that E is the midpoint of DF.
BY MR M. SHABALALA
@NOMBUSO
(3)
(3)
[10]
Page 13
QUESTION 8 (FEB/MARCH 2010)
ΔABC is a right-angled triangle with t
point on AB such that DE
᳸
. D is a point on AC such that BD
AC and E is a
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
Page 14
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.
Prove the theorem that states CAP
ABC
(5)
9.2
RS is a diameter of the circle with centre O. Chord ST is produced to W. Chord SP
produced meets the tangent RW at V. R
5
Calculate the size of:
9.2.1
9.2.2
9.2.3
9.2.4
(1)
WRS
(2)
W
Prove that V
BY MR M. SHABALALA
(3)
PTS
@NOMBUSO
(4)
[15]
Page 15
QUESTION 10 (NOVEMBER 2011)
AB is a diameter of the circle ABCD. OD is drawn parallel to BC and meets AC in E.
[5]
If the radius is 10 cm and AC = 16 cm, calculate the length of ED.
QUESTION 11 (NOVEMBER 2011)
CD is a tangent to circle ABDEF at D. Chord AB is produced to C. Chord BE cuts chord
AD in H and chord FD in G. AC || FD and FE = AB. Let t
11.1
‴ and t
.
(6)
11.2
Determine THREE other angles that are each equal to ‴.
Prove that ΔBHD ||| ΔFED.
(5)
11.3
Hence, or otherwise, prove that AB.BD = FD.BH.
(2)
[13]
BY MR M. SHABALALA
@NOMBUSO
Page 16
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 ACO
Calculate, with reasons, AOD.
BY MR M. SHABALALA
[5]
@NOMBUSO
Page 17
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
Prove the theorem that states PRS
PTS
h
(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
intersects the circle ABD at F. DE, DB and DF are joined.
14.2.1
h. FC
FD
Calculate, with reasons, in terms of ‴:
(3)
(b)
(2)
(a)
14.2.2
th
DEB
Hence, or otherwise, prove ED || BC.
(3)
[13]
BY MR M. SHABALALA
@NOMBUSO
Page 18
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
‴
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
(4)
, show that
[16]
BY MR M. SHABALALA
@NOMBUSO
Page 19
QUESTION 17 (NOVEMBER 2012)
17.1
If in Δ LMN and Δ FGH it is given that
states
17.2
and
, prove the theorem that
.
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 = 2‴ – 10 units.
Calculate the value of ‴.
(4)
[11]
BY MR M. SHABALALA
@NOMBUSO
Page 20
QUESTION 18 (NOVEMBER 2012)
18.1
Complete the following statement:
The angle between the tangent and the chord is equal ...
18.2
(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.
5
᳸
Let
,
h,
and
ཪ, calculate the values of , h, and ཪ
(9)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 21
QUESTION 19 (NOVEMBER 2012)
O is the centre of the circle CAKB.
AK produced intersects circle AOBT at T.
㌠t
‴
19.1
Prove that
19.2
Prove that AC || KB.
19.3
Prove ΔBKT ||| ΔCAT
19.4
If AK : KT = 5 : 2 , determine the value of
h
BY MR M. SHABALALA
(3)
‴
(5)
(3)
AC
KB
@NOMBUSO
(3)
[14]
Page 22
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
(1)
20.2
Determine an expression for DC in terms of ‴ if CM = 4‴ units.
20.3
Hence, or otherwise, calculate the length of the radius.
(4)
Determine an expression for OM in terms of ‴.
(2)
[7]
BY MR M. SHABALALA
@NOMBUSO
Page 23
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.
h and K
Calculate, with reasons, the sizes of the following angles:
21.1
(2)
21.2
(2)
21.3
(2)
21.4
(2)
21.5
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 24
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
area of ΔQRA
support your answer.
BD
BQ
(2)
. Show all working to
(5)
[13]
BY MR M. SHABALALA
@NOMBUSO
Page 25
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
23.1
23.2
23.3
GD.
‴.
Write down, with reasons, FOUR other angles each equal to ‴.
(8)
Prove that BE.DE = AE.FE
(7)
Prove that t
(4)
t
BY MR M. SHABALALA
[19]
@NOMBUSO
Page 26
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
24.2
Calculate the length of ED if AF
(2)
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
Page 27
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.
and
5
Calculate, with reasons, the values of :
26.1
26.2
26.3
(2)
(3)
t㌠
(4)
[9]
BY MR M. SHABALALA
@NOMBUSO
Page 28
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.
Let
27.1
‴
Determine FOUR other angles, each equal to ‴.
(8)
27.3
Prove that KE
(3)
27.4
Prove that
27.2
Prove that
᳸
ET.
‴
(3)
(6)
[20]
BY MR M. SHABALALA
@NOMBUSO
Page 29
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
Calculate the sizes of the following angles:
28.1
(2)
28.2
(2)
28.3
(3)
28.4
(3)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 30
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
‴ and
.
30.1
Name, with reasons, THREE angles each equal to
(6)
30.2
Prove that ΔPRS ||| ΔPST.
(3)
30.3
Prove that PS × RT = RS × PT.
(3)
[12]
BY MR M. SHABALALA
@NOMBUSO
Page 31
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
Page 32
QUESTION 32 (NOVEMBER 2014)
32.1
In the diagram, O is the centre of the circle passing through A, B and C.
㌠ t
h , ㌠ t
‴ and ㌠
Determine, with reasons, the size of:
32.1.1
32.1.2
32.2
(2)
‴
(2)
In the diagram, O is the centre of the circle passing through A, B, C and D. AOD
is a straight line and F is the midpoint of chord CD. t
and OF are joined
Determine, with reasons, the size of:
32.2.1
32.2.2
(2)
t㌠
BY MR M. SHABALALA
(2)
@NOMBUSO
Page 33
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
32.3.1
13, AE
‴ and BC
‴
Give reasons for the statements below.
(2)
32.3.2
Calculate the length of AB.
(4)
[14]
BY MR M. SHABALALA
@NOMBUSO
Page 34
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
AE
and
in ΔADE
DB
(1)
EC
Using the above diagram, complete the proof of the theorem
Construction: Construct altitudes (heights)
(5)
BY MR M. SHABALALA
@NOMBUSO
Page 35
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)
CM
(3)
area ΔFDC
(4)
AM
ME
area ΔBDC
BY MR M. SHABALALA
[16]
@NOMBUSO
Page 36
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
circle, where T is the point of contact. Chord RTP is drawn. Let
34.1
‴ and
Give reasons for the statements below.
Let
‴ and
statement
34.1.1
34.1.2
34.1.3
34.1.4
Reason
‴
‴
WT SP
34.1.5
(5)
BY MR M. SHABALALA
@NOMBUSO
Page 37
WR RP
34.2
Prove that RT
(2)
34.3
Identify, with reasons, another TWO angles equal to .
(4)
34.4
Prove that
(3)
34.5
Prove that ΔRTS ||| ΔRQP.
(3)
34.6
Hence, prove that
(3)
RS
[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
And
PM
OM
5
‴ units, AB
units
35.1
Write down the length of MB.
(1)
35.2
Give a reason why OM
(1)
35.3
Show that OP
35.4
Calculate the value of ‴.
‴
BY MR M. SHABALALA
AB.
units.
(2)
(3)
[7]
@NOMBUSO
Page 38
QUESTION 36 (FEB/MARCH 2015)
In the diagram below, the circle with centre O passes through A, B, C and D. AB DC and
BOC
36.1
. The chords AC and BD intersect at E. EO, BO, CO, and BC are joined.
Calculate the size of the following angles, giving reasons for your answers:
36.1.1
36.1.2
36.2
(2)
t
(2)
36.1.3
(4)
Prove that BEOC is a cyclic quadrilateral.
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 39
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.
and
5
37.3.1
Give a reason why
37.3.2
State, with reasons, TWO other angles equal to 30
37.3.3
Determine, with reasons, the size of:
37.3.4
.
(1)
(3)
(a)
(3)
(b)
(4)
Prove that WR
BY MR M. SHABALALA
RV × RS
@NOMBUSO
(5)
[16]
Page 40
QUESTION 38 (FEB/MARCH 2015)
In ΔTRM,
given that RT
38.1
᳸ . NP is drawn parallel to TR with N on TM and P on RM. It is further
3PN
Give reasons for the statements below
Statement
Reason
In ΔPNM and ΔRTM:
38.1.1
……………………………………………….
is common
38.2
38.1.2
Prove that
38.3
ΔPNM ||| ΔRTM ……………………………………………….
PM
RM
Show that RN
PN
BY MR M. SHABALALA
RP .
@NOMBUSO
(2)
(2)
(4)
[8]
Page 41
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
h cm ,
MT
2 cm, OM
‴ cm.
39.1
(1)
39.2
Write OP in terms of ‴ and a number
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.
40.1
40.2
.
Calculate the size of the following angles, giving reasons for your answers:
40.1.1
(3)
40.1.2
(3)
40.1.3
(3)
If it is further given that NW WR , prove that TNWO is a cyclic quadrilateral
(4)
[13]
BY MR M. SHABALALA
@NOMBUSO
Page 42
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
t
(2)
41.2
Why is SB RP ?
(1)
41.3
Prove, with reasons, that:
41.1.2
41.3.1
41.3.2
41.3.3
(2)
(4)
ΔAPS ||| ΔRWS
RS
(4)
WS AS
(3)
[16]
BY MR M. SHABALALA
@NOMBUSO
Page 43
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
42.2
Prove that ΔPBQ ||| ΔPQA
42.3
PR
(7)
(4)
Prove that the lengths of PA, PR and PB(in this order) form a geometric sequence.(3)
[14]
BY MR M. SHABALALA
@NOMBUSO
Page 44
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.
43.1.2 Use QUESTION 43.1.1 to prove that
43.2
㌠
h
(1)
(3)
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
Page 45
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
‴
‴
44.1
Prove that
(4)
44.2
Prove that AKBT is a cyclic quadrilateral.
(2)
44.3
Prove that TK bisects
(4)
44.4
Prove that TA is tangent to the circle passing through the points A, K and H.
44.5
S is a point in the circle such that the points A, S, K and B are concyclic.
t.
Explain why A, S, B and T are also concyclic.
(2)
(2)
[14]
BY MR M. SHABALALA
@NOMBUSO
Page 46
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.
45.2
E is a point on BC such that BE : EC
(3)
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
(5)
45.2.3
Prove that ΔBAC ||| ΔFEC.
45.2.4
Write down, giving reasons, the radius of the circle passing
Calculate the length of AC.
through points A, B and C.
(4)
(2)
[17]
BY MR M. SHABALALA
@NOMBUSO
Page 47
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
NR
46.2.1
46.2.2
1,5; PM 2; KM
2,5; MN
; MR 1,25 and
0,75.
Prove that ΔKPM ||| ΔRNM.
Determine the length of NQ.
(3)
(6)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 48
QUESTION 47 (FEB/MARCH 2016)
47.1
In the diagram below, tangent KT to the circle at K is parallel to the chord NM.
NT cuts the circle at L. ΔKML is drawn.
and
h .
Determine, giving reasons, the size of :
47.1.1
(2)
47.1.2
(3)
47.1.3
(2)
47.1.4
(2)
47.1.5
(1)
BY MR M. SHABALALA
@NOMBUSO
Page 49
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. t t
h
and t
‴
.
Calculate , giving reasons, the value of ‴.
(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.
48.2
Write down, with reasons, TWO ratios each equal to
48.3
Prove that
48.4
It is further given that ABCD is a rhombus. Prove that ACGF is a cyclic
.
quadrilateral
(1)
t
tt
(4)
(5)
(3)
[13]
BY MR M. SHABALALA
@NOMBUSO
Page 50
QUESTION 49 (FEB/MARCH 2016)
49.1
In the diagram below, ΔABC and ΔPQR are given with
DE is drawn such that AD
49.1.1
PQ and AE
49.1.2
ΔPQR
49.1.3
Hence, prove that
t
BY MR M. SHABALALA
and ㌠
PR.
Prove that ΔADE
Prove that DE BC.
,t
(2).
(3)
㌠
(2)
@NOMBUSO
Page 51
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
49.2.2
Prove that ΔROP ||| ΔRVS.
(4)
Prove that ST
(6)
49.2.3
49.2.4
PS?
(1)
Prove that ΔRVS ||| ΔRST.
BY MR M. SHABALALA
VT TR
@NOMBUSO
(3)
[21]
Page 52
QUESTION 50 (SEPTEMBER 2016)
50.1
In the diagram below, BAED is a cyclic quadrilateral with BA || DE. BE = DE and
t
. The tangent to the circle at D meets AB produced at C
Calculate, with reasons the sizes of the following.
50.1.1
50.1.2
50.1.3
50.1.4
50.1.5
(2)
t
(2)
t
(2)
t
(2)
t
(3)
BY MR M. SHABALALA
[11]
@NOMBUSO
Page 53
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
. PR produced meets QS at B. BC || SP and cuts the
chord QR at D. QR produced meets SP at A. Let t
51.1
‴
(4)
51.2
Name, with reasons, 3 angles equal to ‴
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
(4)
QB.CP
[17]
BY MR M. SHABALALA
@NOMBUSO
Page 54
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
52.1.1
Give a reason why
(1)
52.1.2
Calculate, with reasons, the size of :
(a)
(3)
(b)
(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
and AC
48
52.2.1
Calculate AT.
(1)
52.2.2
If OS
(5)
BY MR M. SHABALALA
5
OT, calculate the radius OA of the circle.
[12]
@NOMBUSO
Page 55
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.
53.1
‴ and ㌠
Give a reason why EACH of the following is true:
53.1.1
53.1.2
t
‴
(1)
(1)
53.2
t
Prove that BCDE is a cyclic quadrilateral
(2)
53.3
Which TWO other angles are each equal to ‴?
(2)
53.4
Prove that t
t㌠t
㌠
BY MR M. SHABALALA
(3)
[9]
@NOMBUSO
Page 56
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
(6)
Page 57
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
‴, KF
‴, ML
and LH
54.2.1
Give a reason why
54.2.2
Prove that :
54.2.3
HG.
(1)
(a)
GH
(3)
(b)
ΔMFH ||| ΔMGF
(5)
(c)
‴
(2)
Show that
BY MR M. SHABALALA
(3)
‴
[20]
@NOMBUSO
Page 58
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,
and
h
Determine, with reasons, the size of the following angles:
55.1
(2)
55.2
(2)
55.3
(2)
55.4
(2)
[8]
BY MR M. SHABALALA
@NOMBUSO
Page 59
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
56.1
3 : 2.
Write down the value of the following ratios:
56.1.1
56.1.2
ST
(2)
SV
(1)
TQ
VR
56.2
Prove that
(4)
56.3
Complete the following statement: ΔVWS ||| Δ …
(1)
56.4
Determine WV : PR
(2)
[10]
BY MR M. SHABALALA
@NOMBUSO
Page 60
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
. The tangent at P meets SQ produced at T. OQ intersects
PS at A.
57.2.1
Give a reason why
(1)
57.2.2
Prove that PQ bisects
57.2.3
Determine
57.2.4
Prove that PT is a tangent to the circle that passes through points
.
in terms of .
P, O and A.
57.2.5
Prove that
BY MR M. SHABALALA
(4)
(2)
(2)
᳸ .
@NOMBUSO
(5)
[19]
Page 61
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.
‴.
58.1
Prove that TPLN is a cyclic quadrilateral.
58.2
Determine, giving reasons, the size of
58.3
Prove that:
58.3.2
ΔKTP ||| ΔKLN
58.3.2
KT KN
BY MR M. SHABALALA
KT
(3)
in terms of ‴.
(3)
(3)
TP
@NOMBUSO
(5)
[14]
Page 62
QUESTION 59 (NOVEMBER 2017)
In the diagram, points A, B, D and C lie on a circle. CE AB with E on AD produced.
Chords CB and AD intersect at F. t
59.1
5
and ㌠
5
Calculate, with reasons, the size of:
59.1.1
59.1.2
59.2
(3)
(2)
㌠
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
Page 63
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
‴, FB
‴, AG
and GC
side AD such that GH CK and GE CD.
60.1
60.2
h . H, E and K are points on
Prove that :
60.1.1
FG
BC
(2)
60.1.2
AH
AE
(3)
HK
ED
It is further given that AH
BY MR M. SHABALALA
5 and ED
12, calculate the length of EK.
@NOMBUSO
(5)
[10]
Page 64
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
61.2
Prove that:
MN
(1)
61.2.1
MN TS
(2)
61.2.2
TMNS is a cyclic quadrilateral
(4)
61.2.3
OS.MN
(5)
2ON.WS
[12]
BY MR M. SHABALALA
@NOMBUSO
Page 65
QUESTION 62 (NOVEMBER 2017)
[5 marks]
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
meets CB produced at A. CD is produced to E such that EA
Let ㌠
62.2.1
‴.
Give a reason why:
t
᳸
(1)
ABDE is a cyclic quadrilateral
(1)
t
‴
(1)
(a)
AD
AE
(3)
(b)
ΔADB ||| ΔACD
(3)
(a)
(b)
(c)
62.2.2
62.2.3
AC
Prove that:
It is further given that BC
t
t
t
t
(a)
Prove that
(b)
Hence, prove that ΔADE is equilateral.
(2)
(4)
[20]
BY MR M. SHABALALA
@NOMBUSO
Page 66
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.
63.1
63.2
Calculate, with reasons, the size of :
63.1.1
(2)
63.1.2
(2)
If it is further given that PQ
TR:
63.2.1
Calculate, with reasons, the size of
63.2.2
Prove that
BY MR M. SHABALALA
TR
TS
RQ
RS
@NOMBUSO
(2)
(2)
[8]
Page 67
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.
‴ and
Determine, with reasons,
in terms of ‴
QUESTION 65(GP)
[6]
ALB is a tangent to circle LMNP. ALB||MP.
Prove that:
65.1
LM
LP
(4)
65.2
LN bisects MNP
(4)
65.3
LM is a tangent to circle MNQ
BY MR M. SHABALALA
(4)
@NOMBUSO
Page 68
QUESTION 66 (FEB/MARCH 2018)
In the diagram, DEFG is a quadrilateral with DE
meet in H. t
45 and GF
80. The diagonals GE and DF
and t
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
Page 69
QUESTION 67 (FEB/MARCH 2018)
[5 marks]
67.1
Proof angle at the centre
× angle at circumference.
67.2
In the diagram, the circle with centre F is drawn. Points A, B, C and D lie on the
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 t
67.2.1
‴.
Determine, with reasons, the size of EACH of the following in
terms of ‴:
(2)
(b)
(2)
(a)
㌠
67.2.2
Prove, with reasons, that AFED is a cyclic quadrilateral.
67.2.3
Prove, with reasons, that
67.2.4
If area ΔAEB
BY MR M. SHABALALA
‴
6, 25 × area ΔDEC, calculate
@NOMBUSO
(4)
(6)
AE
ED
(5)
[24]
Page 70
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
68.3
Calculate OB
(1)
68.4
AB (correct to one decimal place)
(3)
68.5
The radius of the circle CAB.
(2)
ubits, determine OC in terms of .
(1)
[9]
BY MR M. SHABALALA
@NOMBUSO
Page 71
QUESTION 69 (SEPTEMBER 2018)
69.1
In the diagram below, ABCD is a cyclic quadrilateral of the circle with centre O.
Use the diagram to prove the theorem which states that t t
h .
(5)
69.2
KOL is the diameter of the circle KPNML having centre O. R is the point on chord KN,
such that KR = RO. OR is produced to P. Chord KM bisects
and cuts LP in T.
‴
Prove with reasons that:
69.2.1
69.2.2
69.2.3
TK TL
KOTP is a cyclic quadrilateral.
PN MK
BY MR M. SHABALALA
@NOMBUSO
(5)
(3)
(3)
[16]
Page 72
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
5
Calculate with reasons, the following ratios:
AS
70.1
(3)
SC
RT
70.2
(2)
TC
ARS
70.3
(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
71.2
If PA = 6 units, TC = 5 units
and PT = ‴, show that
71.3
71.4
‴
5‴
㌠
(5)
(2)
Calculate the length of PT.
(2)
Calculate the length of PD.
(3)
[12]
BY MR M. SHABALALA
@NOMBUSO
Page 73
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.
Let
.
Calculate, with reasons, the size of EACH of the following angles:
72.1.1
(2)
72.1.2
(2)
72.1.3
(1)
72.1.4
(2)
72.1.5
(3)
BY MR M. SHABALALA
@NOMBUSO
Page 74
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
Page 75
QUESTION 73 (NOVEMBER 2018)
(5 marks)
73.1
Proof for opposite angles in a cyclic quadrilateral
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.
‴ and
73.2.1
.
Name, giving a reason, another angle equal to:
(a)
(b)
(2)
‴
(2)
73.2.2
Prove that SCDB is a cyclic quadrilateral.
73.2.3
It is further given that t
and
Prove that SD is not a diameter of circle BDS.
(3)
.
(4)
[16]
BY MR M. SHABALALA
@NOMBUSO
Page 76
QUESTION 74 (NOVEMBER 2018)
In the diagram, ABCD is a cyclic quadrilateral such that AC
produced to M such that AM
74.1
74.2
MC. Let t
CB and DC
CB. AD is
‴.
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
CM
DC
t
BY MR M. SHABALALA
AM
(6)
AB
tಸ ‴
(2)
[16]
@NOMBUSO
Page 77