Western ape Education Department Examination Preparation Learning Resource 06 GEOMETRY MTHEMTIS Grade Geometry Questions Grade Theorems on Razzia Ebrahim Senior urriculum Planner for Mathematics E-mail: Razzia.Ebrahim@wced.info Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/85- grade-?itemid=- Website: http://wcedeportal.co.za Tel: 0 467 67 ell: 08 708 0448
Index Page. 06 Feb-March Paper 4. 05 November Paper 5 7. 05 June Paper 8 0 4. 05 Feb-March Paper 5. 04 November Paper 4 6 6. 04 Exemplar Paper 7 7. 0 November Paper 8 8. 0 November Paper 9 0 9. 0 November Paper 0. 00 November Paper 4. 009 November Paper 5 6. 008 November Paper 7 8
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Mathematics/P DE/November 05 Give reasons for LL statements in QUESTIONS 8, 9, 0 and. QUESTION 8 8. In the diagram below, cyclic quadrilateral D is drawn in the circle with centre O. O D 8.. omplete 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. () 8.. Use QUESTION 8.. to prove that  + Ĉ = 80. () 5
Mathematics/P DE/November 05 8. In the diagram below, D is a common chord of the two circles. Straight lines DE and F are drawn. hords and EF are drawn. D F E Prove that EF. (5) [9] 6
Mathematics/P DE/November 05 QUESTION 9 In the diagram below, is drawn in the circle. T and T are tangents to the circle. The straight line THK is parallel to with H on and K on. K is drawn. Let  = x. T H K x S 9. Prove that Kˆ = x. (4) 9. Prove that KT is a cyclic quadrilateral. () 9. Prove that TK bisects Kˆ. (4) 9.4 Prove that T is a tangent to the circle passing through the points, K and H. () 9.5 S is a point in the circle such that the points, S, K and are concyclic. Explain why, S, and T are also concyclic. () [4] 7
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Mathematics/P DE/November 04 GIVE RESONS FOR YOUR STTEMENTS IN QUESTIONS 8, 9 ND 0. QUESTION 8 8. In the diagram, O is the centre of the circle passing through, and. Â = 48, Ô x and Ĉ y. 48 O y x Determine, with reasons, the size of: 8.. x () 8.. y () 8. In the diagram, O is the centre of the circle passing through,, and D. OD is a straight line and F is the midpoint of chord D. O Dˆ F 0 and OF are joined. O F 0 D Determine, with reasons, the size of: 8.. Fˆ () 8.. ˆ () 4
Mathematics/P DE/November 04 8. In the diagram, and E are tangents to the circle at and E respectively. is a diameter of the circle. =, E = x and = x + 7. x E x + 7 8.. Give reasons for the statements below. omplete the table on DIGRM SHEET. Statement Reason (a) (b) ˆ 90 x () 8.. alculate the length of. (4) [4] 5
Mathematics/P DE/November 04 QUESTION 0 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. hord RTP is drawn. Let Rˆ 4 x and Rˆ y P S X W x 4 R y T 4 Q Y 0. Give reasons for the statements below. omplete the table on DIGRM SHEET 6. Let Rˆ 4 x and Rˆ Statement 0.. Tˆ = x 0.. Pˆ = x 0.. WT SP 0..4 Ŝ = y y Reason 0..5 Tˆ = y (5) 6
Mathematics/P Grade Exemplar DE/04 Give reasons for your statements in QUESTIONS 8, 9 and 0. QUESTION 8 8. omplete the following statement: The angle between the tangent and the chord at the point of contact is equal to... () 8. In the diagram,,,, D and E are points on the circumference of the circle such that E. E and D produced meet in F. GH is a tangent to the circle at. ˆ = 68 and Fˆ = 0. G F 68 4 E D 0 H Determine the size of each of the following: 8.. 8.. 8.. 8..4 Ê () ˆ () Dˆ () Ê () 8..5 Ĉ () [9] 7
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Mathematics/P DE/November 0 QUESTION 8 8. omplete the following statement: The angle between the tangent and the chord is equal... () 8. 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. M PˆR = 75 P Qˆ T = 9 Q Tˆ R = 4 Let T PˆW = a, R PˆT = b, M PˆQ = c and R Tˆ U = d, calculate the values of a, b, c and d. R Q M 9 75 c P b a 4 T d U W (9) [0] 9
Mathematics/P DE/November 0 QUESTION 0 In the diagram below, O is the centre of the circle. hord is perpendicular to diameter D. M : MD = 4 : 9 and = 4 units. D O M 0. Determine an expression for D in terms of x if M = 4x units. () 0. Determine an expression for OM in terms of x. () 0. Hence, or otherwise, calculate the length of the radius. (4) [7] TOTL: 00 0
Mathematics/P DE/November 0 In the next FOUR questions, ensure you give reasons for each statement you make. QUESTION 8 8. In the diagram below, O is the centre of the circle. PQ is a tangent to the circle at. and are points on the circumference of the circle., and are joined. Prove the theorem that states ÂP = ˆ. O P Q (5) 8. RS is a diameter of the circle with centre O. hord ST is produced to W. hord SP produced meets the tangent RW at V. Rˆ = 50. W V R P T O alculate the size of: S 8.. W Rˆ S () 8.. Ŵ () 8.. Pˆ () 8..4 Prove that Vˆ = PTˆ S. (4) [5]
Mathematics/P DE/November 0 QUESTION 9 is a diameter of the circle D. OD is drawn parallel to and meets in E. O E D If the radius is 0 cm and = 6 cm, calculate the length of ED. [5] QUESTION 0 D is a tangent to circle DEF at D. hord is produced to. hord E cuts chord D in H and chord FD in G. FD and FE =. Let Dˆ = x and Dˆ = y. 4 F H 4 D G E 0. Determine THREE other angles that are each equal to x. (6)
Mathematics/P DE/November 00 QUESTION 7 In the diagram below is a diameter of the circle with centre O. and chord D intersect at E., and D are also chords of the circle. OD is joined. E D. D E O If Ĉ =, calculate, with reasons, the size of: 7. Â () 7. Dˆ () 7. Show that E bisects D Â () [8]
Mathematics/P DE/November 00 QUESTION 8 8. 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. G J O T K Prove the theorem that states H Tˆ = TĴK. (5) 8. ED is a diameter of the circle, with centre O. ED is extended to. is a tangent to the circle at. O intersects E at F. D O. Ê = x. 4 D O F x 8.. Write down, with reasons, THREE other angles equal to x. (4) E 8.. Determine, with reasons, ˆ E in terms of x. () 4
Mathematics/P DoE/November 009 QUESTION 8 8. In the diagram below, O is the centre of the circle. K, L and M are points on the circumference of the circle. M K O L Prove that the obtuse angle at O, K ÔM = KLˆM. (6) 5
Mathematics/P DoE/November 009 8. In the diagram below, O is the centre of the circle. P, Q, R and S are points on the circumference of the circle. TOQ is a straight line such that T lies on PS. PQ = QR and Qˆ = x. Q x P O T R S 8.. alculate, with reasons, P ˆ in terms of x. 8.. Show that TQ bisects P QR. () () QUESTION 9 8.. Show that STOR is a cyclic quadrilateral. () [5] O is the centre of the circle below. OM. The radius of the circle is equal to 5 cm and = 8 cm. O M 9. Write down the size of Ĉ. () 9. alculate: 9.. The length of M, with reasons () 9.. () rea ΔOM : rea Δ 6
Mathematics/P DoE/November 008 QUESTION 7 7. omplete the statements below by filling in the missing word(s) so that the statements are ORRET: 7.. The angle subtended by a chord at the centre of a circle is. () 7.. The angle between the tangent and a chord is. () 7.. The opposite angles of a cyclic quadrilateral are. () 7. In the figure below, RDS is a tangent to circle O at D. If = D and D S = 40, calculate, with reasons, the measures of: 7.. D () 7.. 7.. () () 7..4 O () O 4 5 40 R D S [9] 7
Mathematics/P DoE/November 008 QUESTION 8 In the diagram below, points R, P,, Q and T lie on a circle. R bisects R and = Q. R and TQ produced meet at. R P T Q Prove that: 8. Q bisects PQ () 8. TR = T () 8. P = T R P () [8] 8