2016 2008 CAPS NCS Geometry Gr 12

Western Cape Education Department Examination Preparation Learning Resource 2016 GEOMETRY MATHEMATICS Grade 12 Razzia Ebrahim Senior Curriculum Planner for Mathematics E-mail: Razzia.Ebrahim@wced.info Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835- grade-12?Itemid=-1 Website: http://wcedeportal.co.za Tel: 021 467 2617 Cell: 083 708 0448

Index Page 1. 2016 Feb-March Paper 2 3–5 2. 2015 November Paper 2 6–7 3. 2015 June Paper 2 8–9 4. 2015 Feb-March Paper 2 10 – 11 5. 2014 November Paper 2 12 – 15 6. 2014 Exemplar Paper 2 16 – 17 7. 2013 November Paper 3 18 – 19 8. 2012 November Paper 3 20 – 21 9. 2011 November Paper 3 22 – 23 10. 2010 November Paper 3 24 – 25 11. 2009 November Paper 3 26 – 28 12. 2008 November Paper 3 29 – 30 2

Mathematics/P2 DBE/Feb.—Mar. 2016 NSC QUESTION 9 In the diagram below, EO bisects side AC of AACE. EDO is produced to B such that BO=OD. AD and CD produced meet EC and EA at G and F respectively. A 91 Give a reason why ABCD is a parallelogram. (1) 9.2 Write down, with reasons, TWO ratios each equal to DB" (4) 9.3 Prove that A = F,. (5) 9.4 It is further given that ABCD is a rhombus. Prove that ACGF is a cyclic quadrilateral. (3) [13] Copyright reserved Please turn over

Mathematics/P2 DBE/Feb.—Mar. 2016 NSC QUESTION 10 10.1 In the diagram below, AABC and APQR are given with A=P, B=Q and C=R. DE is drawn such that AD =PQ and AE=PR. 10.1.1 Prove that AADE = APQR. 10.1.2 Prove that DE || BC. 10.1.3 Hence, prove that AB = AC PR’ Copyright reserved Please turn over

Mathematics/P2 10.2 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 NSC VR at T. ST, OP and SV are drawn. 10.2.1 10.2.2 10.2.3 Copyright reserved Why is OP 1 PS? Prove that AROP ||| ARVS. Prove that ARVS || | ARST. Prove that ST?=VT.TR. DBE/Feb.—Mar. 2016 Q) (4) (3) (6) [21]

Mathematics/P2 DBE/November 2015 NSC QUESTION 10 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. B 17 E 8 C F D A 10.1 Calculate, with reasons, the length of DC. (3) 10.2 E is a point on BC such that BE : EC = 3 : 1. EF is parallel to BD with F on DC. 10.2.1 Calculate, with reasons, the length of CF. (3) 10.2.2 Prove that ∆BAC | | | ∆FEC. (5) 10.2.3 Calculate the length of AC. (4) 10.2.4 Write down, giving reasons, the radius of the circle passing through points A, B and C. (2) [17] Copyright reserved Please turn over 6

Mathematics/P2 DBE/November 2015 NSC QUESTION 11 11.1 Complete the following statement: If the sides of two triangles are in the same proportion, then the triangles are ... (1) 11.2 In the diagram below, K, M and N respectively are points on sides PQ, PR and QR of ∆PQR. KP = 1,5; PM = 2; KM = 2,5; MN = 1; MR = 1,25 and NR = 0,75. P 2 1,5 M 2,5 1,25 K 1 R 0,75 N Q 11.2.1 Prove that ∆KPM | | | ∆RNM. (3) 11.2.2 Determine the length of NQ. (6) [10] Copyright reserved 7

Mathematics/P2 DBE/2015 SCE QUESTION 10 10.1 Complete the following statement of the theorem in the ANSWER BOOK: If a line divides two sides of a triangle in the same proportion, then ... (1) 10.2 In the diagram ABC is a triangle with F on AB and E on AC. BC || FE. D ison AF with “- AE = 12 units and EC =8 units. B EF D G A 12 E 8 10.2.1 Prove that DE | | FC. (G) 10.2.2 If AB= 14 units, calculate the length of BF. (6D) [7] Copyright reserved Please turn over

Mathematics/P2 DBE/2015 SCE 112 In the diagram C is the centre of the circle DAP. BA is a tangent to the circle at A. CD is produced to meet the tangent to the circle at B. DP and DA are drawn. E isa point on BA such that EC bisects DCA. Let Cc =x. 11.2.1 Prove that ABAD | | | ABCE. (7) 11.2.2 If itis also given that AB=8 units and AC = 6 units, calculate: (a) The length of BD (5) (b) The length of BE G3) (c) The size of x G) [24] Copyright reserved

Mathematics/P2 NSC DBE/Feb.—Mar. 2015 93 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. R, = 30° and R, =50°. 93.1 93.2 9.3.3 9.3.4 Copyright reserved Give a reason why R, =30°, State, with reasons, TWO other angles equal to 30°. Determine, with reasons, the size of: (a) 8, (bo) Vv Prove that WR?=RV « RS. 10 Please turn over (4) (5) [22]

Mathematics/P2 DBE/Feb.—Mar. 2015 NSC QUESTION 10 In ATRM, M= 90°. NP is drawn parallel to TR with N on TM and P on RM. It is further given that RT = 3PN. T N I ib R PB M 10.1 Give reasons for the statements below. Use DIAGRAM SHEET 5. Statement Reason In APNM and ARTM : 10.1.1] N,=T Miscommon 10.1.2 | APNM ||] ARTM J ooccccccccccsescseseescsceseecscsescseesessessssesesessneaee (2) 1 10.2 Prove that EN =>. 2) RM 3 10.3 Show that RN? — PN? = 2RP?. (4) [8] Copyright reserved ll

Mathematics/P2 DBE/November 2014 NSC QUESTION 9 9.1 In the diagram, points D and E lie on sides AB and AC of ABC respectively such that DE | | BC. DC and BE are joined. A k h D 1 1 E B C 9.1.1 Explain why the areas of DEB and DEC are equal. (1) 9.1.2 Given below is the partially completed proof of the theorem that states AD AE that if in any ABC the line DE | | BC then  . DB EC Using the above diagram, complete the proof of the theorem on DIAGRAM SHEET 4. Construction: Construct the altitudes (heights) h and k in ADE . area ΔADE 1 ADh  2  ........ area DEB 1 BDh 2 area ΔADE AE  ......................  area DEC EC But areaDEB = .............................. (reason: .................................) area ΔADE   ............................... area DEB AD AE   DB EC (5) Copyright reserved Please turn over 12

Mathematics/P2 DBE/November 2014 NSC 9.2 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. A F D G E M B C Calculate, giving reasons, the ratio of: EM 9.2.1 (3) AM CM 9.2.2 (3) ME area FDC 9.2.3 area BDC (4) [16] Copyright reserved Please turn over 13

Mathematics/P2 DBE/November 2014 NSC QUESTION 10 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 R̂ 4  x and R̂ 2  y P 1 2 S 1 2 1 2 T 3 4 X 1 W 2 1 x 2 Q y 3 4 3 2 R 1 Y 10.1 Give reasons for the statements below. Complete the table on DIAGRAM SHEET 6. Let R̂ 4  x and R̂ 2  y Statement Reason 10.1.1 T̂3 = x 10.1.2 P̂1 = x 10.1.3 WT | | SP 10.1.4 Ŝ1 = y 10.1.5 T̂2 = y (5) Copyright reserved Please turn over 14

Mathematics/P2 DBE/November 2014 NSC WR.RP 10.2 Prove that RT  (2) RS 10.3 Identify, with reasons, another TWO angles equal to y. (4) 10.4 Prove that Q̂ 3  Ŵ2 . (3) 10.5 Prove that RTS | | | RQP. (3) WR RS 2 10.6 Hence, prove that  . RQ RP 2 (3) [20] Copyright reserved 15

Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar QUESTION 9 In the diagram, M is the centre of the circle and diameter AB is produced to C. ME is drawn perpendicular to AC such that CDE is a tangent to the circle at D. ME and chord AD intersect at F. MB = 2BC. M B A C 3 1 2 1 2 1 2 x F 3 2 3 4 1 D E 9.1 If D̂ = x, write down, with reasons, TWO other angles each equal to x. (3) 4 9.2 Prove that CM is a tangent at M to the circle passing through M, E and D. (4) 9.3 Prove that FMBD is a cyclic quadrilateral. (3) 9.4 Prove that DC2 = 5BC2. (3) 9.5 Prove that ∆DBC | | | ∆DFM. (4) DM 9.6 Hence, determine the value of . (2) FM [19] Copyright reserved Please turn over 16

Mathematics/P2 DBE/2014 NSC – Grade 12 Exemplar QUESTION 10 10.1 In the diagram, points D and E lie on sides AB and AC respectively of ∆ABC such that DE | | BC. Use Euclidean Geometry methods to prove the theorem which states that AD AE = . DB EC A D E B C (6) 10.2 In the diagram, ADE is a triangle having BC | | ED and AE | | GF. It is also given that AB : BE = 1 : 3, AC = 3 units, EF = 6 units, FD = 3 units and CG = x units. A 3 B C x G E 6 F 3 D Calculate, giving reasons: 10.2.1 The length of CD (3) 10.2.2 The value of x (4) 10.2.3 The length of BC (5) area ΔABC 10.2.4 The value of area ΔGFD (5) [23] Copyright reserved 17

Mathematics/P3 DBE/November 2013 NSC GIVE REASONS FOR YOUR STATEMENTS AND CALCULATIONS IN QUESTIONS 9 TO 12. QUESTION 9 In AADC, 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. A B Cc F E D 9.1 Determine the value of AF : FE (2) 9.2 Calculate the length of ED if AF = 8 cm. (4) [6] Conia iene NN 0 00000 Peetu ove 18

Mathematics/P3 9 DBE/November 2013 NSC QUESTION 12 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 W, =x 12.1 Determine FOUR other angles, each equal to x. (8) 12.2 Prove that T = 90°— x. (3) 12.3 Prove that KE = ET. (3) 12.4 Prove that KE? = OE.WE (6) [20] Cone rere 080 19

Mathematics/P3 DBE/November 2012 NSC NOTE: Give reasons for all statements made in QUESTION 7, QUESTION 8, QUESTION 9 and QUESTION 10. QUESTION 7 7.1 If in ∆ LMN and ∆ FGH it is given that L̂ = F̂ and M̂ = Ĝ , prove the theorem that LM LN states = . FG FH L M N F G H (7) 7.2 In the diagram below, ∆VRK 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. V 4 2x – 10 T P 66 9 9 K R (4) [11] Copyright reserved Please turn over 20

Mathematics/P3 DBE/November 2012 NSC QUESTION 9 O is the centre of the circle CAKB. AK produced intersects circle AOBT at T. AĈB = x C x B 1 T 4 2 3 1 O 2 K 1 2 3 A 9.1 Prove that T̂ = 180° − 2 x . (3) 9.2 Prove AC || KB. (5) 9.3 Prove ∆BKT ||| ∆CAT (3) AC (3) 9.4 If AK : KT = 5 : 2, determine the value of KB [14] Copyright reserved Please turn over 21

Mathematics/P3 DBE/November 2011 NSC QUESTION 9 AB is a diameter of the circle ABCD. OD is drawn parallel to BC and meets AC in E. B O A E C D If the radius is 10 cm and AC = 16 cm, calculate the length of ED. [5] QUESTION 10 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 D̂ 4 = x and D̂ 1 = y . A F B 1 3 2 3 H 2 1 G 3 1 1 2 3 2 2 1 E 4 C D 10.1 Determine THREE other angles that are each equal to x. (6) 10.2 Prove that ΔBHD ||| ΔFED. (5) 10.3 Hence, or otherwise, prove that AB.BD = FD.BH. (2) [13] Copyright reserved Please turn over 22

Mathematics/P3 DBE/November 2011 NSC QUESTION 11 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. B A F E C D P Q 11.1 Show that E is the midpoint of AD. (2) 11.2 Prove PQ || FE. (3) 11.3 If PQ is 60 cm, calculate the length of FE. (5) [10] Copyright reserved 23

Mathematics/P3 DBE/November 2010 NSC 8.2 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 . C 2 D 1 4 B 3 1 2 1 2 O 1 3 2 F x A E 8.2.1 Write down, with reasons, THREE other angles equal to x. (4) 8.2.2 Determine, with reasons, CB̂E in terms of x. (3) 8.2.3 Prove that F is the midpoint of BE. (4) 8.2.4 Prove that ∆CBD ||| ∆CEB. (2) 8.2.5 Prove that 2EF.CB = CE.BD. (3) [21] Copyright reserved Please turn over 24

Mathematics/P3 7 DBE/November 2010 NSC QUESTION 9 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. D 8 x A E C y 8 B If DE = x units and AB = y units, calculate x and y. [6] QUESTION 10 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. F E M A || || D B C 10.1 Prove MDCE is a cyclic quadrilateral. (3) 10.2 Prove that MC2 = MB2 + DC2 – DB2. (3) 10.3 Calculate CE if AB = 60 mm, ME = 40 mm and BC = 20 mm. (4) [10] Copyright reserved 25

Mathematics/P3 DoE/November 2009 NSC QUESTION 9 O is the centre of the circle below. OM ⊥ AC. The radius of the circle is equal to 5 cm and BC = 8 cm. O B A M C 9.1 Write down the size of BĈA. (1) 9.2 Calculate: 9.2.1 The length of AM, with reasons (3) 9.2.2 Area ΔAOM : Area ΔABC (3) Copyright reserved Please turn over 26

Mathematics/P3 9 DoE/November 2009 NSC QUESTION 10 AB In the figure below, GB || FC and BE || CD. AC = 6 cm and = 2. BC A G H B F E C D 10.1 Calculate with reasons: 10.1.1 AH : ED (4) BE 10.1.2 CD (2) 10.2 If HE = 2 cm, calculate the value of AD × HE. (2) [8] Copyright reserved Please turn over 27

Mathematics/P3 10 DoE/November 2009 NSC QUESTION 11 In the figure below, AB is a tangent to the circle with centre O. AC = AO and BA || CE. DC produced, cuts tangent BA at B. 1 A E 2 2 3 4 1 2 1 F 3 4 B 2 3 O 1 C 2 1 D 11.1 Show Ĉ2 = D̂1 . (3) 11.2 Prove that ΔACF ||| ΔADC. (3) 11.3 Prove that AD = 4AF. (4) [10] Copyright reserved 28

Mathematics/P3 DoE/November 2008 NSC QUESTION 9 In the figure below, PQ is a diameter to circle PWRQ. SP is a tangent to the circle at P. ∧ Let P1 = x P Q 1 x 2 2 1 W 2 1 1 2 1 T 3 2 R S ∧ 9.1 Why is P R Q = 90°? (1) ∧ ∧ 9.2 Prove that P1 = S . (3) 9.3 Prove that SRWT is a cylic quadrilateral. (3) 9.4 Prove that ∆QWR /// ∆QST. (3) 9.5 If QW = 5 cm, TW = 3 cm, QR = 4 cm and WR = 2 cm, calculate the length of: 9.5.1 TS (3) 9.5.2 SR (3) [16] Copyright reserved Please turn over 29

Mathematics/P3 DoE/November 2008 NSC QUESTION 10 In the figure below, ∆ABC has D and E on BC. BD = 6 cm and DC = 9 cm. AT : TC = 2 : 1 and AD || TE. A T F C E D B CE 10.1 Write down the numerical value of (1) ED 10.2 Show that D is the midpoint of BE. (2) 10.3 If FD = 2 cm, calculate the length of TE. (2) 10.4 Calculate the numerical value of: Area of ∆ADC 104.1 (1) Area of ∆ABD Area of ∆TEC 10.4.2 (3) Area of ∆ABC [9] Copyright reserved 30