Grade 11 Euclidean Geometry 4. Circles 4.1 Terminology

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Grade 11 Euclidean Geometry 2014GRADE 11 EUCLIDEAN angentTERMINOLOGYAn arc is a part of the circumference of a circleA chord is a straight line joining the ends of an arc.A radius is any straight line from the centre of the circle to a point on thecircumferenceA diameter is a special chord that passes through the centre of the circle. ADiameter is the length of a straight line segment from one point on thecircumference to another point on the circumference, that passes through thecentre of the circle.A segment is the part of the circle that is cut off by a chord. A chord divides acircle into two segmentsA tangent is a line that makes contact with a circle at one point on thecircumference (AB is a tangent to the circle at point P).1

Grade 11 Euclidean Geometry 20144.2 SUMMARY OF THEOREMS4.2.1Definitions2

Grade 11 Euclidean Geometry 20144.2.2Chords and Midpoints3

Grade 11 Euclidean Geometry 20144.2.3 Angles in circles4

Grade 11 Euclidean Geometry 20144.2.4Cyclic Quadrilaterals5

Grade 11 Euclidean Geometry 20144.2.4Tangents6

Grade 11 Euclidean Geometry 20147

Grade 11 Euclidean Geometry 20144.3 PROOF OF THEOREMSAll SEVEN theorems listed in the CAPS document must be proved. However, there are fourtheorems whose proofs are examinable (according to the Examination Guidelines 2014) ingrade 12. In this guide, only FOUR examinable theorems are proved. These four theoremsare written in bold.1. The line drawn from the centre of a circle perpendicular to the chord bisects thechord.2. The perpendicular bisector of a chord passes through the centre of the circle.3. The angle subtended by an arc at the centre of a circle is double the angle subtendedby the same arc at the circle (on the same side of the arc as the centre).4. Angles subtended by an arc or chord of the circle on the same side of the chord are equal.5. The opposite angles of a cyclic quadrilateral are supplementary.6. Two tangents drawn to a circle from the same point outside the circle are equal in length(If two tangents to a circle are drawn from a point outside the circle, the distancesbetween this point and the points of contact are equal).7. The angle between the tangent of a circle and the chord drawn from the point ofcontact is equal to the angle in the alternate segment.The above theorems and their converses, where they exist, are used to prove riders.8

Grade 11 Euclidean Geometry 20149

Grade 11 Euclidean Geometry 2014ORTheorem 1The line drawn from the centre of a circle, perpendicular to a chord, bisects the chord.10

Grade 11 Euclidean Geometry 201411

Grade 11 Euclidean Geometry 2014ORTheorem 4The angle subtended by an arc at the centre of a circle is double the size of the anglesubtended by the same arc at the circumference of the circle.12

Grade 11 Euclidean Geometry 2014Theorem 7The opposite angles of a cyclic quadrilateral are supplementary.13

Grade 11 Euclidean Geometry 2014Theorem 10The angle between a tangent and a chord, drawn at the point of contact, is equal to the anglewhich the chord subtends in the alternate segment.14

Grade 11 Euclidean Geometry 20144.4ACTIVITIESGEOMETRY 11. The sketch alongside shows chord BD cutting AE at C. A is thecentre of the circle and AE BD. If EC 3cm and BD 14cm,calculate the area of the circle.ECBDA2. The sketch shows circle centre O with OC ‖ AB . OCˆ B 76º and  x.Calculate x.O76A15xBoC

Grade 11 Euclidean Geometry 2014CONDITIONS FOR QUADRILATERAL TO BE CYCLICIfORopp. int.anglessuppl.Then PQRS is acyclic quad.IfORThen PQRS is acyclic quad.IfThen PQRS is acyclic quad.16Angles inthe sameseg.ext. angleequal to int.opp. angle

Grade 11 Euclidean Geometry 20143. The diagram shows circles with centres Q and O, and MTˆR 400.MT and RT are not necessarily tangents to the smaller ˆ O3.4P̂17Oo40T

Grade 11 Euclidean Geometry 20144.In the accompanying figure, AB is a diameter of the circle with centre O. DC is atangent to the circle at point C. Chord AC is drawn. D is a point on the tangent DC sothat Aˆ1 Aˆ 2 .AD21O2121CBEProve that:4.1 AD ‖ OC4.2 ADˆ C 90 18

Grade 11 Euclidean Geometry 20145.In the figure PS is a diameter of the circle with centre T. BQ is a tangent to the circleand TR is perpendicular to QS. RTˆS x.B12Q3 41R2P12x12TS5.1 Prove that TR ‖ PQ.5.2 Determine, with reasons, other four angles each equal to x.5.3 Prove that TQRS is a cyclic quadrilateral.19

Grade 11 Euclidean Geometry 20146.In the figure below, diagonals AC and BD of cyclic quadrilateral ABCDintersect at P such that AP PB. FPG is a tangent to circleFD2A1212 1P34122B1CGABP.Prove that:6.1 FG ‖ DC6.220

Grade 11 Euclidean Geometry 20147. The sketch below shows circles BKAC and KMTB intersecting at K and B, andABˆ T 90 . AB and BT are not diameters, BT is not a tangent to the smaller circle, andAB is not a tangent to the larger circle.B31 2C1A22313K24121M21S7.1 Prove that SABT is a cyclic quadrilateral.7.2 Express in terms of .7.3 Prove that21T

Grade 11 Euclidean Geometry 2014SOLUTIONSGEOMETRY 1E1. BC CD 7cm . AC BDLet AC xCBThen the radius r x 3DAD² AC² CD² PythagThus (x 3)² x² 7²A x² 6x 9 x² 49 6x 40 x 6⅔ r 9⅔ Area (9⅔) ² 293, 56 cm²2. ABˆ C 104º . Co-int ’s;OC ‖ AB. Reflex AOˆ C 2B 208º Obtuse AOˆ C 152ºO76 x 28º co-int ’s;OC ‖ AB.AxB22oC

Grade 11 Euclidean Geometry 20143.MP12341Q121R3.1 Q̂ 2 140º . opposite T̂ in cyclic quad MQRT3.2 Ô1 80º . 2 T̂ on the circumference3.3 PMˆ R 90º . in a semi-circleM 3 50º . sum of isosceles OMR PMˆ O 14003.4 P̂ 70º . Q̂ 2 is the exterior of isosceles QMP2323Oo40T

Grade 11 Euclidean Geometry 2014A4.D21O2121CBE4.1C2 A1 . OA OC (radii) C 2 A 2 . A 1 A2(given) AD ‖ OC . alternate angles equal4.2 OĈE 90º . tangent CE radius OC A Dˆ C 90 . corresponding angles; AD ‖ OC5.B5.1 Q 2 3 90º . in semi-circle TR ‖ PQ . corresponding ’s equal5.2 T 2 x . TR is a line of symmetry of isos TQS12 P x . ½ T at the centreQ3 41R2 Q 2 x . PT TQ (radii)PQ 4 P x . tan BQR ; chord QS1T242x12S

Grade 11 Euclidean Geometry 20145.3 Q 4 RTˆS x. Thus TQRS is cyclic . chord RS6.FD2A1212 1P34122B1CG6.1) P4 A1 . tang PG; chord PBBut P4 P1 . vert opp angles and A1 D1 . chord BC P1 D1 FG ‖ DC alt ’s equal6.2 P4 A1 . tang PG; chord PBP2 B1 . tang FP; chord APBut A1 B1 . equal chords AP and BP P4 P2 P1 FP bisects AP̂D25

Grade 11 Euclidean Geometry 20147.B31 2C1A22313K24121M21S7.1M1 B 3 . exterior of cyclic quad BKMTLet M 1 B 3 x ABˆ K 90º – x Ĉ . ’s in the same segment ASˆM 90º . sum CSM BAST is cyclic . ABˆ T ASˆM 180 7.2 Let T2 y S 2 y . same chord AB S1 90º – S 2 90º – T27.3M 1 K 4 T 1 . ext of 26T

Grade 11 Euclidean Geometry 2014But M 1 B 3 , K 4 K 1 (vert opp ’s) and T 1 B 1 (chord AS) Bˆ3 Kˆ 1 Bˆ127

Grade 11 Euclidean Geometry 2014MORE ACTIVITIES28

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Grade 11 Euclidean Geometry 2014 8 4.3 PROOF OF THEOREMS All SEVEN theorems listed in the CAPS document must be proved. However, there are four theorems whose proofs are examinable (according to the Examination Guidelines 2014) in grade 12. In this guide, only FOUR examinable theorems are proved. These four theorems are written in bold. 1.