A Guide to Circle GeometryTeaching ApproachIn Paper 2, Euclidean Geometry should comprise 35 marks of a total of 150 in Grade 11 and40 out of 150 in Grade 12. This section of Mathematics requires both rote learning as well ascontinuous practice. Pen and paper repetition is the best way to get this right. Each pupilshould have a correct handwritten copy of every theorem to refer to and to memorise. Thetheorems and their proofs, as well as the statement of the converses, must be learned forexamination purposes. The theorems, converses, and other axioms must be used to solveriders and should also be used in formal proofs. In these cases the correct (andunderstandable) abbreviation of a theorem or its corollary can be used.The emphasis during assessment will be on the correctness of formal arguments inexaminations and notation will be scrutinised carefully. In most cases the student needs tofollow the statement, reason, conclusion format. Please refer to the task answers to ensurecorrect setting out is followed. The numbering of theorems is author dependent and thereasons ‘Theorem 3’ will not be acceptable; neither will reasons like ‘bow tie’ or ‘wind surfer’.All the results and definitions from previous grades are acceptable as axioms and do notneed to be proved for the circle geometry results.The proofs of the theorems should be introduced only after a number of numerical and literalriders have been completed and the learners are comfortable with the application of thetheory. When attempting a rider, it is a good idea to use colour to denote angles which areequal as well as cyclic quads, tangents etc. This will assist the learner in making the visualconnections. An alternate method is to let one angle be equal to a variable, for example x,and continue around the diagram until all required angles are known. In all cases teachersmust ensure that sketches and diagrams are legible. The skill we are aiming toward is thenfor the learner to write up your findings in a formal manner.“The more I practise, the luckier I get” is Gary Player’s quote for golf, but very applicable tothis section of the syllabus.
Video SummariesSome videos have a ‘PAUSE’ moment, at which point the teacher or learner can choose topause the video and try to answer the question posed or calculate the answer to the problemunder discussion. Once the video starts again, the answer to the question or the rightanswer to the calculation is given.Mindset suggests a number of ways to use the video lessons. These include: Watch or show a lesson as an introduction to a lesson Watch of show a lesson after a lesson, as a summary or as a way of adding in someinteresting real-life applications or practical aspects Design a worksheet or set of questions about one video lesson. Then ask learners towatch a video related to the lesson and to complete the worksheet or questions,either in groups or individually Worksheets and questions based on video lessons can be used as shortassessments or exercises Ask learners to watch a particular video lesson for homework (in the school library oron the website, depending on how the material is available) as preparation for thenext days lesson; if desired, learners can be given specific questions to answer inpreparation for the next day’s lesson1. Introducing Circle GeometryIn this video we cover three topics: Firstly the origins and uses of Euclidian Geometryand more specifically circle geometry; secondly the concept of a formal proof and theimportance thereof and lastly, the terminology relating to a circle.2. Chords and RadiiThis video introduces the theorem relating to chords and radii. It is followed up with someexamples of where the theory is applied.3. Angles at CentreThis video covers the theorem dealing with perpendicular bisector of the chord, themeaning of the word ‘subtends’ and the theorem dealing with the angle at the centre ofthe circle. It concludes by applying the theorems to some examples.4. Chords Subtending AnglesIn this video we introduce the theorem which considers the angles subtended by chordsat the circumference and then apply it to some examples.5. Interior Angles in Cyclic QuadrilateralsIn this video we introduce the idea of a cyclic quadrilateral; one theorem related to it andthen we apply the theorem to some examples.6. Exterior Angles in Cyclic QuadrilateralsThe theorem dealing with the opposite interior angles of a cyclic quadrilateral isdiscussed and some examples are worked through.
7. Proving Cyclic QuadrilateralsIn this video we look at different ways of proving a quadrilateral is a cyclic quadrilateral.That means proving that all four of the vertices of a quadrilateral lie on the circumferenceof a circle.8. Tangents from a PointThe axiom that a tangent and a radius at the point of contact are always perpendicular isdiscussed and then this is used to prove that the tangents from the same point are equalin length.9. The Tan-Chord TheoremThe tan-chord theorem is discussed in this lesson.10. The Converse Tan-Chord TheoremThis video deals with the converse of the tan-Chord theorem and an examination stylequestion is worked through.11. Working with Circle GeometryA problem which combines a number of bits of theory is dealt with and then thepresenter reflects on how to approach problems in an examination.Resource ometry/11-geometry-01.cnxmlplusA summary of pre-Grade 10 etry/segments-andangles/intro euclid/v/euclid-as-the-father-ofgeometryKhan Academy gives video clips on all sectionsof geometryhttp://www.youtube.com/watch?v KvaD0x8QdDAYou tube presentation on chords and theirperpendicular bisectorshttp://www.youtube.com/watch?v XRk5zhXzhWAYou tube presentation on inscribed and xmlplusText and video clips on all the terminologyneeded for this section, the formal proofs oftheorems and worked try/Geometry%20Part%201.pdfSummary of geometry done to this point and thefirst few theorems and ry/Geometry%20Part%202.pdfNotes on angles subtended as well as ry/Circle%20Geometry%20Part%203.pdfNotes on cyclic quads and examples
e%20Geometry%20Part%204.pdfNotes on circle geometry - exampleshttp://www.youtube.com/watch?v TiIq5i80JA4You Tube presentation of the tan-chord theorem
TaskQuestion 1AC and BD are diameters. AB // CD and AC and meets BD at O.numerical magnitude of the angles labelled x, y and z.Find theQuestion 2A, B and C are points on the circumference of the circle centre O. Find the numerical sizesof the angles labelled x and y.Question 3AOB is a diameter, BC 20mm, AC // DB. Find the magnitudes of the angles labelled x and yand the lengths AC and OB.
Question 4PT is a tangent to the circle at S. Prove that SA // CB.Question 5In the diagram126.96.36.199and. Prove:Question 6is a tangent to the circle centre O,following three angles:andand.Determine, in terms of p, the
Question 7CD is a tangent to circle ABDEF at D. Chord AB is produced to C. Chord BE cuts chord AD inH and chord FD in G. AC // FD and FE AB. Letand7.1 Determine THREE other angles that are each equal to x.7.2 Prove that7.3 Hence, or otherwise, prove that AB BD FD BH
Task Answers:Question 1L’s at centre 2X L’s at circum.Alt L’s AB//DCL’s at centre, reflexQuestion 2L’s at centre 2X L’s at circumis isosRadii OB OCL’s inQuestion 3Diam subtends R L’sAlt L’s AC//DBL’s inAC 20mmIsosPythagmmQuestion 4Tan chordIsosSA//CBRadii OC OBAlt L’s equalQuestion 5Alt L’s AE//BC5.1Ext L’s cyclic quadExt L’s cyclic quadButAlt L’s AE//BC5.2proved aboveBE//DCCorresp L’s
oppo L’s cyclic quad sup5.3co-int L’s supandaboveQuestion 6L’s at centreext L’stangentradiusQuestion 77.1tan chord theoremtan chord theorem or L’s in same segmentAlt L’s, CA//DF7.2 InandL’s in same segment chds subt L’s///LLL/// ’s7.3butgivenAB.BD FD.BH
AcknowledgementsMindset Learn Executive HeadContent Manager Classroom ResourcesContent Coordinator Classroom ResourcesContent AdministratorContent DeveloperContent ReviewersDylan BusaJenny LamontHelen RobertsonAgness MunthaliIan L. AtteridgeRonald JacobsProduced for Mindset Learn by TrafficFacilities ManagerFacilities CoordinatorProduction ManagerEditorPresenterStudio CrewGraphicsEmma WilcockCezanne ScheepersBelinda RenneyTumelo MalokaKatleho SerobeAbram TjaleJames TselapediWilson MthembuJenny van der LeijWayne SandersonCreditsPhoto /3/33/Water tower cropped.jpg/105pxWater tower cropped.jpgPhoto by Jim Gordon, http://en.wikipedia.org/wiki/File:Khan As%27ad Pacha Al-%27Azem.jpgThis resource is licensed under a Attribution-Share Alike 2.5 South Africa licence. When using thisresource please attribute Mindset as indicated athttp://www.mindset.co.za/creativecommons
1. Introducing Circle Geometry In this video we cover three topics: Firstly the origins and uses of Euclidian Geometry and more specifically circle geometry; secondly the concept of a formal proof and the imp