SUBJECT: MATHEMATICS CONTENT: EUCLIDEAN GEOMETRY ACTIVITY .

SUBJECT: MATHEMATICSCONTENT: EUCLIDEAN GEOMETRYACTIVITY BOOKLEARNERTERM 1EUCLIDEAN GEOMETRY:JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

CONTENTSPAGETOPIC 1: Euclidean Geometry Grade 11 Content(Mixed Theorems and Applications with Riders)TOPIC 2: Euclidean Geometry Mixed Exercises (Grade 11-Grade 12)(Mixed Theorems and Applications with Riders)ICON DESCRIPTIONMIND IESTERMINOLOGYWORKED EXAMPLESSTEPS2JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

TOPIC: Euclidean GeometryDuration: 9Outcomes: At the end of the session learners must demonstrate an understanding of:1. The following examinable proofs of theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord; The angle subtended by an arc at the centre of a circle is double the size of the angle subtendedby the same arc at the circle (on the same side of the chord as the centre); The opposite angles of a cyclic quadrilateral are supplementary; The angle between the tangent to a circle and the chord drawn from the point of contact is equalto the angle in the alternate segment; A line drawn parallel to one side of a triangle divides the other two sides proportionally; Equiangular triangles are similar.2. Corollaries derived from the theorems and axioms are necessary in solving riders: Angles in a semi-circle Equal chords subtend equal angles at the circumference Equal chords subtend equal angles at the centre In equal circles, equal chords subtend equal angles at the circumference In equal circles, equal chords subtend equal angles at the centre. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of thequadrilateral. If the exterior angle of a quadrilateral is equal to the interior opposite angle of the quadrilateral,then the quadrilateral is cyclic. Tangents drawn from a common point outside the circle are equal in length.3. The theory of quadrilaterals will be integrated into questions in the examination.4. Concurrency theory is excluded.3JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 11.1In the diagram below, O is the centre of the circle. J, K and L are points on the circumference ofthe circle.JKOLˆ 2 JKLˆ .Prove that the obtuse angle at O, JOL1.2ˆ 600.Given circle with centre O. DT TB and ABDAODBTC1.2.1ˆ .Determine TBC1.2.2Show that OD BC .4JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 2In the diagram below, points Q, H, J and K lie on a circle. RK bisects K̂ and RH RP.KR and JH produced meet at P .K1 400.2112Prove that:2.1)ˆ .RH bisects GHP2.2)JK JP.2.3)ˆ .Qˆ JKQQUESTION 33.1 In the diagram alongside, which isreproduced on the diagram sheet,O is the centre of the circle throughA, B and P.Prove the theorem which states thatˆ 2.APBˆAOB5JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

3.2.1 ST is a diameter of the circle.OS PN, TO bisects ST̂P .Prove that3.2.1PUNK is a cyclic quadrilateral3.2.2SO is a tangent to circle KUST3.2.3POST is a cyclic quadrilateralQuestion 44.1It is given that 𝐵𝑂̂𝐷 126 , where 𝑂 is thecentre of the circle, and that 𝐴𝐵 𝐸𝐹.ADEO126 FCB4.1.1 Determine the value of 𝐷𝐶̂ 𝐵.4.1.2 Prove that CDEF is a cyclic quadrilateral.4.2PPOQ is a diameter of the circle andSQR is the tangent to the circle at Q.28 OGiven that 𝑃̂ 28 , determine the value of 𝑅̂ .2112T61 2SQJENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12R

QUESTION 55.1In the diagram below, P, M, T and R are points on a circle having centre O.PR produced meets MS at S. Radii OM and OR and the chords MT and TR are drawn. T 1 148 , PMO 18 and S 43 .Calculate, with reasons, the size of: 5.1.1P5.1.2O15.1.3OMS5.1.4R 3 , if it is given that TMS 6 7JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

5.2In the diagram below, the circle passes through A, B and E. ABCD is a parallelogram. BC is a tangent to the circle at B. AE AB. Let C1 x . 5.2.1Give a reason why B1 x .5.2.2Name, with reasons, THREE other angles equal in size to x.5.2.3Prove that ABED is a cyclic quadrilateral.8JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 6In the diagram below, DA and DB are tangents to the circle at A and B. AF FB.AB produced cuts the line through D, which is parallel to FB, at C. AF produced meets DC at E and DAE x .6.1Find, with reasons, 5 angles each equal to x.6.2Prove that ABED is a cyclic quadrilateral.6.3Prove that ABE 3 DAE .6.4Prove that AD BC. 9JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 77.1In the diagram below, ,andare points on a circle with centre . Use thediagram sheet to prove the theorem that states:“Angles subtended by a chord (or arc) at the circumference on the same side of thechord are equal.”SPO.RQ7.2In the diagram below ,, ,andare points on a circle.̂.Calculate, with reasons, the size of:a)̂b) ̂10JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 8In the figureandLet ̂andare tangents to the given circle.such that ̂is a point onis a point on the circumference,̂ . SQ is drawn.OOOProve that:8.18.28.3is a cyclic quadrilateral.bisectŝ .11JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 99.1C is the centre of the circle passing through A, B, D and E. CB DE and BAˆ D 40 .Calculate with reasons the size of:9.1.1Ĉ19.1.2B̂29.1.3Ĉ 212JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1010.1̂ 1 17 and L̂2 51 .In the diagram below, O is the centre of circle KLNM. MPNQ is a tangent to the circle at N.PL1 2151 2N3OQ2117 12K12MCalculate, giving reasons, the size �1N13JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

10.2In the diagram below, O is the centre of circle MPQ. MQ is extended to R and PR isproduced. MP RP and QP QR.RQ1Ox21312M10.2.1̂ 1 in terms of 𝑥 if R̂ 𝑥.Determine O10.2.2Prove that RP is a tangent to the circle.21P14JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1111.1In the diagram alongside, M is the centre of circle PQRS. PM DC , QR PRand R 2 280Determine, giving reasons, the size of the following angles: 11.1.1 S 2 11.1.2 P S R 11.1.3 Q 11.1.4 P 315JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

11.2In the diagram below, PQ is a tangent to circle SRQWT at Q.PRS is a straight line.RW cuts SQ and QT at K and L respectively.ˆ x.PS QT , RS TW and Q2SR3T21K42W1L3 1212 34QP11.2.1Find , with reasons, three other angles equal to x.Prove that :1111.2.2.2.3ˆ 1 Lˆ 3RPRKQ is a cyclic quadrilateral.16JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1212 .1In the diagram below, P, M, T and R are points on a circle havingcentre O. PR produced meets MS at S. Radii OM and OR andthe chords MT and TR are drawn. T̂1 148o , PM̂O 18o andŜ 43 o.Calculate, with reasons the the size of:12.1.1P̂12.1.2Ô112.1.3OM̂S12.1.4R̂3 if it is given that TM̂S 6o.JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 1217

12.2In the diagram below, the circle passes through A, B and E.ABCD is a parallelogram. BC is a tangent to the circle at B.AE AB. Let Ĉ1 x.12.2.1Give a reason why B̂1 x.12.2.2Name, with reasons, THREE other angles equal to x.12.2.3Prove that ABED is a cyclic quadrilateral.18JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

12.4 In the diagram below PR is a tangent to the circle at Q, OP // TQ and S, U, Q and T are points onthe circle. QS and OP intersect at W. O is the centre of the circle.QP121U 32T2 3W14213R41O2S12.4.1Prove that W is the midpoint of QS.12.4.2Prove that12.4.3Prove that SOQP is a cyclic quadrilateral.ˆˆ.Q 1 Q 219JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1313.1In the figure, KL QR, and pointsM and N on QR are chosen so thatKN PR and LM PQ.PK 3 units, PL 4 units, LR 6 unitsand MN 1,8 units.13.1.1Calculate KQ13.1.2State why QM KL13.1.3Prove that QM NR13.2 In the figure, two circles intersect at A and B. AB produced to M bisectsQÂR . Tangents MQ and MR meet the circles at Q and R such thatQBR is a straight line. AQ and AR are joined.Prove:13.2.1 MQA MBQ13.2.2MR2 AM.MB13.2.3BM.AB QB.BR20JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1414.1 Use the diagram below to prove the theorem that states that the line drawnparallel to one side of a triangle divides the other two sides proportionally.i.e. Given that DE BC prove:AD AE DB EC14.2 In DEF, GH // EF and KH // GF. DK 80 units and KG 120 unitsDetermine, giving reasons,14.2.1DHin simplest fraction formHF14.2.2the length of DE14.2.3Area DHKArea DGFJENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 1221

QUESTION 1515.1P is the centre of the circle with radius 73 units.M is the midpoint of chord QR, N is a point onPR so that PN 40 units. MN PR.15.1.1 Give a reason why PM QR.15.1.2Determine the length of MR (to the nearest whole number),giving reasons.15.2 In the diagram, O is the centre ofthe circle with diameter AOB.The tangent through C intersectsAD produced at F.OD AC and CF AFProve that:15.2.1 FCD CAB15.2.2 FC .CB 2 FD.AEˆ15.2.3 AC bisects FAB22JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1616.1 Complete the statement: The angle between the tangent and a chord drawn to the point ofcontact is equal to .16.2 In the figure O is the centre of the circle.DE is a tangent to the circle at C.DE//AB and COˆ B 144 ADO144 3422C1B1Giving reasons, find the value of :16.2.1 𝐶116.2.2 𝐵2EP16.3 In the given sketch, MN is a diameter of the circle.MPNR is a cyclic quadrilateral and PQ MN.M1 2T3142 1212N124S 3Prove:RQ16.3.1 TSRN is a cyclic quadrilateral16.3.2 MP is a tangent to the circle through PTN16.4 The figure alongside is reproducedon your diagram sheet.Show your constructions onthe diagram sheet and provethe Proportional Division Theorem which states:A line drawn parallel to one side of atriangle divides the other two sides proportionally.ADBEC23JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 1717.1In the diagram below, MVT and AKF are drawn such that M A , V K and T F .Use the diagram in the answer book to prove the theorem which statesthat if two triangles are equiangular, then the corresponding sides areMV MT in proportion, that is AK AF .24JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 18In the accompanying diagram, PS bisects RQ. T is the midpoint of PS and 𝑀𝑇𝑊// 𝑃𝑄.Calculate, with reasons, the numerical value of the following:18.118.2𝑅𝑀𝑅𝑃𝐴𝑟𝑒𝑎 𝑅𝑃𝑆𝐴𝑟𝑒𝑎 𝑅𝑀𝑊QUESTION 1919.1In the diagram below, DEF and PQR are two triangles such that̂ 𝑃̂, 𝐸̂ 𝑄̂ and 𝐹̂ 𝑅̂𝐷DPQEProve the theorem, which states that:RF𝐷𝐸𝑃𝑄 𝐷𝐹𝑃𝑅25JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 2020.1O is the centre of the circle, and ST is a tangent to the circle at T.ˆ.ˆ QUse the diagram to prove the theorem which states that STP26JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

20.2 CD and CE are produced to A and B respectively so that AE is a tangent to theˆ 32 and CDEˆ 63 .circle and AB AE. AED20.2.1.Calculate, giving reasons, the size ofa)b)ĈˆAEB20.2.2 Prove that ABED is a cyclic quadrilateral.20.2.3 Prove that AB is a tangent to the circle through B, D and C.ˆ .20.2.4. Calculate, giving reasons, the size of BDE27JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 21In ABC, D is the midpoint of AB, CD EF and21.1Determine, with reasons, the value of21.2Find the value ofAE 2 .EC 3AF.FBArea ΔBCE(no reasons required).Area ΔFEA28JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

QUESTION 2222.1ˆ P,ˆ and Cˆ Rˆ .ˆ Bˆ QIn the diagram below, ABC and PQR are given with A Line XY is drawn so that AX PQ and AY PR.Use the diagram to prove22.1.1 XY BC22.1.2AB AC PQ PR29JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12

Question 23X1D321B1224 53123C12Y21AIn the diagram XBA is the tangent to the circle at X.- XDY is a chord, with DB constructed so that XB DB.- C is a point on the circle, with YCB perpendicular to XBA.- DCA is a straight line.23.1Prove that Ĉ5 ̂X1 ̂X2.23.2Hence, prove that XBCD is a cyclic quadrilateral.23.3Show that the area of 𝐴𝑋𝑌 1/2 𝑋𝑌 𝐴𝐷 .30JENN: LEARNER MANUAL EUCLIDEAN GEOMETRY GRADE 12