It's about maths ,basically about calculus and Euclidean geometry for those studying that, Exercises of Mathematics

Download It's about maths ,basically about calculus and Euclidean geometry for those studying that and more Exercises Mathematics in PDF only on Docsity! MATHEMATICS GRADE 12 2021 REVISION MATERIAL EUCLIDEAN GEOMETRY A collection of questions from previous question papers (2018 to 2020). Prepared by T Faya I LOVE MATHS SERIES BOOK 2 Take note: Coding of questions. KZN J18 – KZN June 2018 KZN S18 – KZN Sept 2018 ECS18 – Eastern Cape Sept 2018 NM17 – National March 2017 And so on……………. This booklet is divided into 4 SECTIONS. 1. SECTION A – Questions on proofs of examinable theorems. 2. SECTION B – Questions on mostly level 1 and 2 problems. 3. SECTION C – Questions on mostly level 2 to 3 problems. 4. SECTION D – Mostly level 3 to level 4 questions 2 Examinable theorems. 1.1. In the diagram O is the centre of the circle. Use the diagram to prove the theorem which states that if OM ⟘ AB then AM = MB. (6) 1.2 In the diagram below, O is the centre of the circle ABCD. Prove the theorem that states that o180D̂B̂  (6) B B A C D  O O B A M O SECTION A 5  Mostly Level 1 and 2 Questions on: - Calculating angles. - Finding angles equal to a specific angle. - Finding angles equal to a given variable (usually x). EC S18 SECTION B 6 QUESTION 9 FS S18 9.1 In the diagram, PAL is a tangent to the circle at A. C and B are points on the circle such that CB||PAL. AB, AC and PB are drawn. PB intersects AC at D. ?̂?1 = 78o. Determine the sizes of each of the following angels, with reasons: 9.1.1 ?̂? (1) 9.1.2 𝐴?̂?𝐶 (2) 9.1.3 ?̂?3 (1) 9.1.4 ?̂?2 (1) GP S18 QUESTION 8 In the diagram below, TAP is a tangent to circle ABCDE at A. AE || BC and DC = DE. TAE =40° and AEB=60°. 8.1 Identify TWO cyclic quadrilaterals. (2) 8.2 Determine, with reasons, the size of the following angles: 8.2.1 B, (2) 8.2.2 B, (2) 8.2.3 DB (2) 8.2.4 E, (3) NM18 10.2 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 Bex. 10.2.1 Determine, with reasons, the size of EACH of the following in terms of x: (a) Ff, (2) {b) € (2) 10.2.2 Prove, with reasons, that AFED is a cyclic quadrilateral. (4) 10.2.3 Prove, with reasons, that B, =x, (6) 10 NJ18 QUESTION 8 8.1 In the diagram, O is the centre of the circle. Radii OH and OJ are drawn. A tangent is drawn from K_ to touch the circle at H. AHGJ is drawn such that GJ|| KH. 0, = 2x. K 8.1.1 Name, giving reasons, THREE angles, each equal to x. (5) 8.1.2 Prove that H, = Hy. GB) 11 NJ18 8.2 In the diagram, KLMN is a cyclic quadrilateral with KLM =87°. Diagonals LN and MK are drawn. P is a point on the circle and MP is produced to T, a point outside the circle. Chord LP is drawn. LMK = y and N, =2y. 8.2.1 Name, giving a reason, another angle equal to y. Q) 8.2.2 Calculate, giving reasons, the size of: (a) y (3) (b) TPL @) [15] 12 KZNM19 5.2. Inthe sketch below, circle centre O has a tangent KLM. Diameter NQ produced meet the tangent in K. N,=32°and N, =23°. Calculate, with reasons, the size of: 5.2.1 BP, () 5.2.2 POO (2) 5.2.3 L, (2) 5.2.4 NLOQ QQ) 525 -, (2) 5.2.6 PLIK (2) [4] 15 KZNJ19 QUESTION 8 In the figure, TQSW is a cyclic quadrilateral with tangent PR touching the circle at Q. WQ bisects PWR. Q, =x Ww 8.1 Name with reasons 5 other angles each equal to x. (5) 8.2 Prove that: 8.2.1 TS#/PR Q) 8.2.2 Q,=P @) 8.2.3 A TQS is an isosceles triangle (4) 8.2.4 WOP=T, QB) [17] 16 17 QUESTION 8 KZNS19 8.1 In the diagram, ABCD is a cyclic quadrilateral in the circle centered at O. ED is a tangent to the circle at D. Chord AB is produced to F. Radii OC and OD are drawn.  40ED̂A and  65Ĉ2 , Determine, giving reasons, the size of each of the following angles: 8.1.1 2D̂ (3) 8.1.2 CB̂F (4) 20 QUESTION 8 FSS19 DF is a tangent to the circle at E. EKHG is a cyclic quadrilateral. .35FÊK o O is the centre of the circle. EHOK and HK.EK  8.1 Determine, with reasons, the size of each of the following: 8.1.1 4Ê (3) 8.1.2 HK̂E (2) 8.1.3 Ĝ (2) 8.1.4 1Ô (2) 8.2 It is further given that EH = 24 units. KM = 4 units and the radius of the circle EKHG is .x Determine the value of .x (4) [13] ● D E G O K H 1 2 3 4 2 3 M 1 F GPS19 QUESTION 8 In the diagram below, A, B, C and D are points on a circle having contre O, PHT is a tangent to the circle at B, Reflex BOC = 6, = 310° as shown in the diagram below. T, Calculate, giving reasons, the size of: 8.1 Dd, (3) 8.2 B, (2) 83 B,, ifitis given that A= 60°. (4) [9] 21 LPS19 QUESTION 8 In the diagram the vertices of A PNR lie on the circle with centre O. Diameter SR and chord NP intersect at T. OT NP and R, =30°. 8.1 8.2 811 § (3) 812 R, (3) 8.1.3 N, (2) If it is further given that NW = WR, prove that TNWO is a cyclic quadrilateral. (4) [12] 22 25 NN19 QUESTION 8 8.1 In the diagram, PQRS is a cyclic quadrilateral. Chord RS is produced to T. K is a point on RS and W is a point on the circle such that QRKW is a parallelogram. PS and QW intersect at U. 136TŜP and  001Q̂1 . Determine, with reasons, the size of: 8.1.1 R̂ (2) 8.1.2 P̂ (2) 8.1.3 WQ̂P (3) 8.1.4 2Û (2) T P Q W S K R 136° 100° 1 1 1 2 2 2 U 2 1 3 26 8.2 In the diagram, the diagonals of quadrilateral CDEF intersect at T. EF = 9 units, DC = 18 units, ET = 7 units, TC = 10 units, FT = 5 units and TD = 14 units. Prove, with reasons, that: 8.2.1 DĈEDF̂E  (4) 8.2.2 CÊDCF̂D  (3) [16] F D E C T 18 7 10 14 5 9 KZNM20 QUESTION 5 In the diagram below, M is the centre of the circle DABC. EDF is a tangent to the circle at D and FBG is another tangent to the circle at B. : Calculate the following angles, with reasons: 5.1 5.2 5.3 5.4 27 (2) (3) (2) {10} GPS20 QUESTION 8 In the diagram below, points A, B, C and D lie on the circumference of a circle with AD || EGC. CB is produced to E. GD is a tangent to the circle at D and DB = AD. EBA = 67°. G 8.1 81.1 ADC (2) 8.12 a ‘ 813 A ) 81.4 Dd, (3) $1.5 BDG @Q) 8.2 Prove that AB = CD. Q) {11} 30 LPS20 QUESTION 8 8.1 M is the centre of the circle SVQR and SV and QR are equal chords. RP and QP are tangents to the circle at R and Q respectively. RPQ =70°. Calculate, giving reasons, the size of: 81.1 R, (4) 8.12 0, (3) 813 M, (3) 31 MPS20 QUESTION 8 In the diagram, TR is a chord of circle PQRST. QAT.LPAS. QAT 1 PAS and P= §,. 2 In the following questions, give a reason for each statement, 8.1 Name three angles with reasons, each equal to 60°. 4 8.2 Calculate the size of QRS. (2) 8.3 Prove that PS!OR. (2) 8.4 Prove that TR is a diameter of the circle. G3) (11) 32 8.2 In AABC, F and G are points on sides AB and AC respectively. D is a point on GC such that D,=B. 8.2.1 If AF is a tangent to the circle passing through points F, G and D, then prove, giving reasons, that FG || BC. (4) om ‘ AF 2 8.2.2 If it is further given that ¥ps? AC =2x-—6 and GC=x+9, then calculate the value of x. (4) 35 36  Mostly level 2 and level 3 questions. KZNS18 SECTION C 9.2 KOL is the diameter of the cirele KPNML having centre O. R is the point on chord KN. such that KR=RO. OR is produced to P. Chord KM bisects LKN and cuts LP in T. Ki =x. Prove with reasons that: 92.1 TK=TL 9.2.2 KOTP isa cyclic quadrilateral. 92.3 PN // ME 37 (G3) 3) [16] GP S18 8.3 In the diagram below, radius CO is produced to bisect chord AB at D. CA =34mm and AB=40mm Calculate the size of C. (4) 40 GP S18 QUESTION 9 In the diagram below, O is the centre of the circle. ABCD is a cyclic quadrilateral. BA and CD are produced to intersect at E such that AB = AE = AC. Determine in terms of x: 91 B, 2) 9.2 E (6) 93 Cc, G) 94 IfE= CG =x, prove that ED is a diameter of circle AED. ty 41 MP S18 8.2 8.2.1 8.2.2 In the diagram, M is the centre of circle and diameter CMPD is perpendicular tochord AB. AB = 4, PD = ¢ and CP = 15cm. Give a reason why AP = 21. (1) If it is further given that ACAP ||] ABDP, calculate: a) The value of ¢. (4) b) The length of the radius of the circle. (2) 42  MATHEMATICS GRADE 12 2021 REVISION MATERIAL EUCLIDEAN GEOMETRY A collection of questions from previous question papers (2018 to 2020). Prepared by T Faya I LOVE MATHS SERIES BOOK 2 Take note: Coding of questions. KZN J18 – KZN June 2018 KZN S18 – KZN Sept 2018 ECS18 – Eastern Cape Sept 2018 NM17 – National March 2017 And so on……………. This booklet is divided into 4 SECTIONS. 1. SECTION A – Questions on proofs of examinable theorems. 2. SECTION B – Questions on mostly level 1 and 2 problems. 3. SECTION C – Questions on mostly level 2 to 3 problems. 4. SECTION D – Mostly level 3 to level 4 questions 2 Examinable theorems. 1.1. In the diagram O is the centre of the circle. Use the diagram to prove the theorem which states that if OM ⟘ AB then AM = MB. (6) 1.2 In the diagram below, O is the centre of the circle ABCD. Prove the theorem that states that o180D̂B̂  (6) B B A C D  O O B A M O SECTION A 5  Mostly Level 1 and 2 Questions on: - Calculating angles. - Finding angles equal to a specific angle. - Finding angles equal to a given variable (usually x). EC S18 SECTION B 6 QUESTION 9 FS S18 9.1 In the diagram, PAL is a tangent to the circle at A. C and B are points on the circle such that CB||PAL. AB, AC and PB are drawn. PB intersects AC at D. ?̂?1 = 78o. Determine the sizes of each of the following angels, with reasons: 9.1.1 ?̂? (1) 9.1.2 𝐴?̂?𝐶 (2) 9.1.3 ?̂?3 (1) 9.1.4 ?̂?2 (1) GP S18 QUESTION 8 In the diagram below, TAP is a tangent to circle ABCDE at A. AE || BC and DC = DE. TAE =40° and AEB=60°. 8.1 Identify TWO cyclic quadrilaterals. (2) 8.2 Determine, with reasons, the size of the following angles: 8.2.1 B, (2) 8.2.2 B, (2) 8.2.3 DB (2) 8.2.4 E, (3) NM18 10.2 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 Bex. 10.2.1 Determine, with reasons, the size of EACH of the following in terms of x: (a) Ff, (2) {b) € (2) 10.2.2 Prove, with reasons, that AFED is a cyclic quadrilateral. (4) 10.2.3 Prove, with reasons, that B, =x, (6) 10 NJ18 QUESTION 8 8.1 In the diagram, O is the centre of the circle. Radii OH and OJ are drawn. A tangent is drawn from K_ to touch the circle at H. AHGJ is drawn such that GJ|| KH. 0, = 2x. K 8.1.1 Name, giving reasons, THREE angles, each equal to x. (5) 8.1.2 Prove that H, = Hy. GB) 11 NJ18 8.2 In the diagram, KLMN is a cyclic quadrilateral with KLM =87°. Diagonals LN and MK are drawn. P is a point on the circle and MP is produced to T, a point outside the circle. Chord LP is drawn. LMK = y and N, =2y. 8.2.1 Name, giving a reason, another angle equal to y. Q) 8.2.2 Calculate, giving reasons, the size of: (a) y (3) (b) TPL @) [15] 12 KZNM19 5.2. Inthe sketch below, circle centre O has a tangent KLM. Diameter NQ produced meet the tangent in K. N,=32°and N, =23°. Calculate, with reasons, the size of: 5.2.1 BP, () 5.2.2 POO (2) 5.2.3 L, (2) 5.2.4 NLOQ QQ) 525 -, (2) 5.2.6 PLIK (2) [4] 15 KZNJ19 QUESTION 8 In the figure, TQSW is a cyclic quadrilateral with tangent PR touching the circle at Q. WQ bisects PWR. Q, =x Ww 8.1 Name with reasons 5 other angles each equal to x. (5) 8.2 Prove that: 8.2.1 TS#/PR Q) 8.2.2 Q,=P @) 8.2.3 A TQS is an isosceles triangle (4) 8.2.4 WOP=T, QB) [17] 16 17 QUESTION 8 KZNS19 8.1 In the diagram, ABCD is a cyclic quadrilateral in the circle centered at O. ED is a tangent to the circle at D. Chord AB is produced to F. Radii OC and OD are drawn.  40ED̂A and  65Ĉ2 , Determine, giving reasons, the size of each of the following angles: 8.1.1 2D̂ (3) 8.1.2 CB̂F (4) 20 QUESTION 8 FSS19 DF is a tangent to the circle at E. EKHG is a cyclic quadrilateral. .35FÊK o O is the centre of the circle. EHOK and HK.EK  8.1 Determine, with reasons, the size of each of the following: 8.1.1 4Ê (3) 8.1.2 HK̂E (2) 8.1.3 Ĝ (2) 8.1.4 1Ô (2) 8.2 It is further given that EH = 24 units. KM = 4 units and the radius of the circle EKHG is .x Determine the value of .x (4) [13] ● D E G O K H 1 2 3 4 2 3 M 1 F GPS19 QUESTION 8 In the diagram below, A, B, C and D are points on a circle having contre O, PHT is a tangent to the circle at B, Reflex BOC = 6, = 310° as shown in the diagram below. T, Calculate, giving reasons, the size of: 8.1 Dd, (3) 8.2 B, (2) 83 B,, ifitis given that A= 60°. (4) [9] 21 LPS19 QUESTION 8 In the diagram the vertices of A PNR lie on the circle with centre O. Diameter SR and chord NP intersect at T. OT NP and R, =30°. 8.1 8.2 811 § (3) 812 R, (3) 8.1.3 N, (2) If it is further given that NW = WR, prove that TNWO is a cyclic quadrilateral. (4) [12] 22 25 NN19 QUESTION 8 8.1 In the diagram, PQRS is a cyclic quadrilateral. Chord RS is produced to T. K is a point on RS and W is a point on the circle such that QRKW is a parallelogram. PS and QW intersect at U. 136TŜP and  001Q̂1 . Determine, with reasons, the size of: 8.1.1 R̂ (2) 8.1.2 P̂ (2) 8.1.3 WQ̂P (3) 8.1.4 2Û (2) T P Q W S K R 136° 100° 1 1 1 2 2 2 U 2 1 3 26 8.2 In the diagram, the diagonals of quadrilateral CDEF intersect at T. EF = 9 units, DC = 18 units, ET = 7 units, TC = 10 units, FT = 5 units and TD = 14 units. Prove, with reasons, that: 8.2.1 DĈEDF̂E  (4) 8.2.2 CÊDCF̂D  (3) [16] F D E C T 18 7 10 14 5 9 KZNM20 QUESTION 5 In the diagram below, M is the centre of the circle DABC. EDF is a tangent to the circle at D and FBG is another tangent to the circle at B. : Calculate the following angles, with reasons: 5.1 5.2 5.3 5.4 27 (2) (3) (2) {10} GPS20 QUESTION 8 In the diagram below, points A, B, C and D lie on the circumference of a circle with AD || EGC. CB is produced to E. GD is a tangent to the circle at D and DB = AD. EBA = 67°. G 8.1 81.1 ADC (2) 8.12 a ‘ 813 A ) 81.4 Dd, (3) $1.5 BDG @Q) 8.2 Prove that AB = CD. Q) {11} 30 LPS20 QUESTION 8 8.1 M is the centre of the circle SVQR and SV and QR are equal chords. RP and QP are tangents to the circle at R and Q respectively. RPQ =70°. Calculate, giving reasons, the size of: 81.1 R, (4) 8.12 0, (3) 813 M, (3) 31 MPS20 QUESTION 8 In the diagram, TR is a chord of circle PQRST. QAT.LPAS. QAT 1 PAS and P= §,. 2 In the following questions, give a reason for each statement, 8.1 Name three angles with reasons, each equal to 60°. 4 8.2 Calculate the size of QRS. (2) 8.3 Prove that PS!OR. (2) 8.4 Prove that TR is a diameter of the circle. G3) (11) 32 8.2 In AABC, F and G are points on sides AB and AC respectively. D is a point on GC such that D,=B. 8.2.1 If AF is a tangent to the circle passing through points F, G and D, then prove, giving reasons, that FG || BC. (4) om ‘ AF 2 8.2.2 If it is further given that ¥ps? AC =2x-—6 and GC=x+9, then calculate the value of x. (4) 35 36  Mostly level 2 and level 3 questions. KZNS18 SECTION C 9.2 KOL is the diameter of the cirele KPNML having centre O. R is the point on chord KN. such that KR=RO. OR is produced to P. Chord KM bisects LKN and cuts LP in T. Ki =x. Prove with reasons that: 92.1 TK=TL 9.2.2 KOTP isa cyclic quadrilateral. 92.3 PN // ME 37 (G3) 3) [16] GP S18 8.3 In the diagram below, radius CO is produced to bisect chord AB at D. CA =34mm and AB=40mm Calculate the size of C. (4) 40 GP S18 QUESTION 9 In the diagram below, O is the centre of the circle. ABCD is a cyclic quadrilateral. BA and CD are produced to intersect at E such that AB = AE = AC. Determine in terms of x: 91 B, 2) 9.2 E (6) 93 Cc, G) 94 IfE= CG =x, prove that ED is a diameter of circle AED. ty 41 MP S18 8.2 8.2.1 8.2.2 In the diagram, M is the centre of circle and diameter CMPD is perpendicular tochord AB. AB = 4, PD = ¢ and CP = 15cm. Give a reason why AP = 21. (1) If it is further given that ACAP ||] ABDP, calculate: a) The value of ¢. (4) b) The length of the radius of the circle. (2) 42 Nw S18 QUESTION 9 Two equal circles cut each other in A and C, BA and BC are tangents to one circle at A and C respectively and they are chords of the other circle. Eis a point on the circumference of one circle and AE produced cuts the other cirele in D. Chords AE and CD are equal. Prove that: 9.1 c, =¢, (4) 9.2 €, =A, (3) 9.3 E is the centre of the circle that passes through A, C and D. 4) 9.4 AECD. is equilateral. (2) [13] 45 NM18 QUESTION 8 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 8S. SPT=x and S=y. Determine, with reasons, y in terms of x. [6] 46 KZNM19 QUESTION 4 In the diagram below, BC =CE: E, =x and D,=D,. A OY E B D Cc 4.1 Name, with reasons, TWO other angles each equal to x and show that FD = FE. (4) 42 Prove that BF bisects CBA. (4 4.3 Hence. or otherwise. prove that 4, = CBA. (4) [12] 47 LPM19 6.3 In the figure O is the centre of the circle. MCT is a tangent touching the circle at C, OM//DC. M 6.3.1 Give the correct reasons for the following statements: STATEMENT REASON (a) | €, =90° (b) | 7K, =90° (Cc) |. BK=KC d) | 646-5 (e) | But D=0, | 26-646, (8) .. BOCM is a cyclic quad. (3) 6.3.2 Prove ABOK /// AMCK 6.3.3 Determine BO if OK =1 unitand MK =9 units. (5) 50 ECS19 8.3 In the diagram below, PRQ is a tangent to the circle SUR at R. SU. SR and UR are drawn. Lines from S and U produced meet at T outside the circle. R, =x; 8, =y andSTU=x+y 5 Prove that STUR is a cyclic quadrilateral. (5) 8.4 The diagram below shows a circle centre M passing through the points R, P, Q and T. RT is the diameter. PQ is a chord such that PQ || RT and MN 1 PQ. PQ = 16 units, MN =6 units. R P vy M Q T Determine the length of RT. (4) 51 52 QUESTION 9 FSS19 9.1 A circle with centre O is given below. Lines CD and AF are produced to E. x2DÔA  and BD is the diameter. AC||FD. 9.1.1 Determine, with reasons, four other angles that are each equal to x . (6) 9.1.2 Express Ê in terms of x . (2) 9.1.3 Prove that AODE is a cyclic quadrilateral. (2) 9.2 It is further given that ED ∶ DC = 8 ∶ 12 and FE = 10. Calculate the length of AF. (3) [13] ● A B C D E F O 1 2 3 1 2 1 2 3 2x 55 NJ19 8.2 In ABC in the diagram, K is a point on AB such that AK : KB = 3 : 2. N and M are points on AC such that KN || BM. BM intersects KC at L. AM : MC = 10 : 23. Determine, with reasons, the ratio of: 8.2.1 AM AN (2) 8.2.2 LK CL (3) [13] QUESTION 9 NJ19 In the diagram, tangents are drawn from point M outside the circle, to touch the circle at B and N. The straight line from B passing through the centre of the circle meets MN produced in A. NM is produced to K such that BM = MK. BK and BN are drawn. Let x K̂  . 9.1 Determine, with reasons, the size of 1N̂ in terms of x. (6) 9.2 Prove that BA is a tangent to the circle passing through K, B and N. (5) [11] B A N M C L K A 56 QUESTION 9 NN19 In the diagram, O is the centre of the circle. ST is a tangent to the circle at T. M and P are points on the circle such that TM = MP. OT, OP and TP are drawn. Let x1Ô . Prove, with reasons, that x 4 1 MT̂S  . [7] S T M O P 1 1 1 2 2 2 3 x KZNM20 QUESTION 6 In the diagram below, NP and NM are tangents to the circle at P and M. K is a point on the circle KP, KM and PM are chords so that K = x. Lisa point on KM so that KP//LN, L and P are joined. N P 6.1 Prove that NMLP is a cyclic quadrilateral. (4) 6.2 Prove that AKLP is isosceles. (6) [10] 57 ECS20 QUESTION 8 8.1 Complete the following theorem statement: The line drawn from the centre of a circle perpendicular to a chord ... (1) 8.2 In the diagram below, circle ABC with centre O is given. OB = 8 units and AB = BC = 10 units. E is the midpoint of AC. Let OE = y and AE = x, A Cc B Calculate, with reasons, the length of OE. (5) 8.3. Complete the following theorem statement: The angle subtended by an arc at the centre of a circle is ... at the circle (on the same (1) side of the chord as the centre). 60 ECS20 8.4 In the diagram, O is the centre of a circle ABCD. AOD is the diameter and OC is a radius. AB, BC, CD, AC and BD are straight lines. co 1/2 1 D KY Write down, with reasons, an equation that expresses the relationship between each of the given groups of angles. ANGLES EQUATION / REASON RELATIONSHIP e.g. M,;P M, =2xP | Z atcentre=2X Z at | circum. 4.1 | Oy: 8, 84.2 | Dy; Cs; D, 8.4.3 | B; Ba; D,; D, 844 | DG (8) 61 ECS20 QUESTION 9 9.1 Given that PC is a tangent to the circle ACB; BAP and BCQ are straight lines. PC = PQand CPQ = 20°. Prove, stating reasons, that BC is NOT a diameter. 62 (5) GPS20 QUESTION 7 In the figure below, KM is a vertical flag post set in the centre of two circles which lie on the same horizontal plane. MKN = MLK = x°®. The radius of the inner circle ML = runits and the radius of the outer circle MN = 2r units. kK TA Calculate the value of x. Tz If r =5m and LMN = 110°, calculate the length of LN. 65 oe LPS20 8.2. TV and VU are tangents to the circle with centre O at T and U respectively. TSRUY are points on the circle such that RT is the diameter. X is the midpoint of chord TU. T; =y. Prove that: 8.2.1 RU|SY (5) 8.22 f= “ y (5) 66 MPS20 QUESTION 9 M is the centre of the circle. M, =x and a, =y. ZR=ZQ. R be * SN P NY Q 91 Prove that x= 4y. (5) 9.2 If it is further given that B, =R, calculate the value of x and y (6) (14) 67 70 FS S18 10.2.3 Calculate the numerical value of: 10.2.3(a) ABD of Area ADC of Area   (2) 10.2.3(b) ΔABC of Area ΔTEC of Area (3) [16] 10.2 In the figure below, 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 and TE||AD and AT:TC = 2:1 10.2.1 Show that D is the midpoint of BE. (3) 10.2.2 If FD = 2 cm, calculate the length of TE. (3) A B D E C T F 71 QUESTION 11 FS S18 In the diagram, TPR is a triangle with TP = 4,5 units. Q and S are points on TR and PR respectively. QR = 9,6 units, QS = 4 units, TS = 3,6 units, PS = 1,5 units and SR = 12 units. 11.1 Prove that PT is a tangent to the circle which passes through the points T, S and R. (7) 11.2 Calculate the length of TQ. (5) [12] GP S18 P R S Q T 9, 6 12 1, 5 4, 5 3, 6 4 GP S18 QUESTION 11 Tn the diagram below, NPQR is a cyclic quadrilateral with S a point on chord PR. N and S are joined and RNS = PNQ =x. Prove that: 11.1 ANSR ||| ANPQ GB) 11.2 ANQR ||| ANPS @) 11.3 NR.PQ + NP.QR =NQ.PR (4) [10] 72 NW S18 QUESTION 10 10.1 Complete the following theorem to make the statement TRUE: Ifa line divides two sides of a triangle in the same proportion, then the line is... (1) 10.2 Two cireles with centres P and S touch each other externally at C. SP produced intersects circle P at B. A common tangent at R and Q meets SB produced at T. Prove that: 10.2.1 PQ||SR (4) T 1022 Te=2Q(BP+SR) (4) QR 10.2.3 ATOP ||| ATRS GQ) 2 2 —2TP. . 1024 TSs?-cs? = (TP* +BP ott BP cosS) .CS © [18] 75 WCS18 10.2 In AABD. BA is produced to F. AC is drawn with C on BD such that A, = A, = x. CE||DA where E is on AB. 10.2.1 Give two more angles equal to x. (3) 10.2.2 Prove that 2 = 4. (4) DC AC 76 WCS18 QUESTION 11 In the diagram, a circle with tangent CD is drawn. A, B, D and E are points on the circumference of the circle. AE = AB and AB is parallel to ED. Ar=x 11.1 11.2 113 11.4 Give, with reasons, three more angles equal to x. Prove that ADEA ||| ADBC Prove that BC.ED = AE?. Calculate the value of x if it is given that A, = 75°. 17 NJ18 QUESTION 10 In the diagram, FBOE is a diameter of a circle with centre O. Chord EC produced meets line BA at A, outside the circle. D is the midpoint of CE. OD and FC are drawn. AFBC is a cyclic quadrilateral. 10.1 Prove, giving reasons, that: 10.1.1 FC || OD (5) 10.1.2 DOE=BAE 4) 10.1.3 AB x OF = AE x OD (7) 10.2 If it is further given that AT =3TO, prove that SCE? = 2BE.FE (5) [21] 80 NN18 8.2 In the diagram, AAGH is drawn. F and C are pointson AG and AH respectively such that AF = 20 units, FG = 15 units and CH = 21 units. D isa 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. G 8.2.1 Explain why FC || GH. (1) 8.2.2 Calculate, with reasons, the length of DM. (5) 81 NNI8 QUESTION 10 In the diagram, ABCD is a cyclic quadrilateral such that AC 1 CB and DC =CB. AD is produced to M such that AM 1 MC. Let Bex. M Cc 10.1 Prove that: 10.1.1 MC isa tangent to the circle at C (5) 10.1.2 AACB ||| ACMD G3) 10.2 Hence, or otherwise, prove that: 2 10.2.1 = = AM (6) pc? AB 10.2.2 ~ =sin’? x (2) [16] 82 85 E D H F G KZNS19 10.2 In the diagram, DE is a tangent to the circle at E and DFG is a secant intersecting the circle at F and G. DE = EF = FG. H is a point on EG such that FH || DE. 10.2.1 Determine, giving reasons, 3 angles each equal to FÊD . (4) 10.2.2 Prove that: a) ∆DEF ||| ∆DGE (3) b) D̂ = 072 . (5) 10.2.3 If it is further given that DF = k units and FG = 2 units, prove that 422  kk . (3) 10.2.4 Determine, giving reasons, the ratio of GE GH in terms of k. (2) [24] 1 2 2 1 3 ECS19 QUESTION 9 In the diagram below, ABCD is a parallelogram. AD and AC are produced to E and F respectively so that EF || DC. AF and DB intersect at O. AD = 12 units: DE = 3 units: DC = 14 units: CF = 5 units. A yg oO. ay y, “ \ A DZ ff A ie. \ E< > \ 1 9.1 Calculate, giving reasons, the length of: 9.1.1 AC @) 91.2 AO qd) 913 EF @) 92 Prove that “*°"°" sare 5 areaAAEF 25 G3) [10] 86 ECS19 10.2 In the diagram below, O is the centre of a semi-circle ACB. S is a point on the circumference and T lies on AC such that STO 1 AB. Diameter AB is produced to P, such that PC is a tangent to the semi-circle at C. Let C, = x. 10.2.1 Write down, with reasons, 2 other angles equal to x. 10.2.2. Prove that ATOC [ll ABPC 10.2.3. Prove that TO.PC = OB.BP 10.2.4 If BP = OB, show that 30C? = PC? 87 G3) G) 2) @ [19] GPS19 QUESTION 11 In the diagram below, the circle with centre O is drawn. OQ is drawn parallel to a tangent to the circle at D. ER is drawn with § on OQ. RD is produced to P and PQ is joined, PE=xunits, PQ=x+9units, PD= ox unitsand =DO-x+3 units. x+9o=s [1.1 Calculate the length of RO. (4) 11.2 If OS = 1,4 units and § is the midpoint of ER, determine the length of DE. (2) 113 If the area of APED= 2,7 units’, find the area of APER. (4) [10] 90 LPS19 9.2 Inthe diagram QR of A PQR is produced such that XS |] YR. Q8 Calculate, giving reasons, the value of RS 91 (6) LPS19 10.2 P, A, Q, Rand § lie on the circle with centre O. SB touches the circle at S and RW = WP. AS and RP are straight lines. Prove that: 10.2.1 SB||RP (5) 10.2.2 A APS ||| ARWS (4) 10.2.3 RS? = WS.AS (4) 2 1024 as=R¥_.ws (4) Ws 92 95 NJ19 10.2 In the diagram, O is the centre of the circle and CG is a tangent to the circle at G. The straight line from C passing through O cuts the circle at A and B. Diameter DOE is perpendicular to CA. GE and CA intersect at F. Chords DG, BG and AG are drawn. 10.2.1 Prove that: (a) DGFO is a cyclic quadrilateral (3) (b) GC = CF (5) 10.2.2 If it is further given that CO = 11 units and DE = 14 units, calculate: (a) The length of BC (3) (b) The length of CG (5) (c) The size of .Ê (4) A B C E F O 1 1 1 2 3 4 2 G 2 D 96 NN19 10.2 In the diagram, ST and VT are tangents to the circle at S and V respectively. R is a point on the circle and W is a point on chord RS such that WT is parallel to RV. SV and WV are drawn. WT intersects SV at K. Let x2Ŝ . 10.2.1 Write down, with reasons, THREE other angles EACH equal to x. (6) 10.2.2 Prove, with reasons, that: (a) WSTV is a cyclic quadrilateral (2) (b) WRV is isosceles (4) (c) WRV ||| TSV (3) (d) TS KV SR RV  (4) [25] S 1 1 1 1 1 2 2 2 2 2 3 3 3 4 x W K R V T QUESTION 8 In the diagram below, TP is a diameter in the circle with centre O. TP is extended to R. RM is. a tangent to the circle at N. MO intersects chord NT at W. NP//MO. WT0=30° M 8.1 Give, with reasons, THREE other angles each equal to 30°, . (3) 8.2 Determine RNT (2) 8.3° Prove that: 8.3.1 ARNP /// ARTN (3) 8.3.2 TW. RN=>RT.NP (5) [13] 97