# 11X1 T13 01 definitions & chord theorems (2011)

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• 1. Circle Geometry

2. Circle GeometryCircle Geometry Definitions 3. Circle GeometryCircle Geometry Definitions 4. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to thecircumference radius 5. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to thecircumferenceDiameter: an interval passing through the radius centre, joining any two pointson the circumferencediameter 6. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to thecircumferenceDiameter: an interval passing through the radius centre, joining any two pointson the circumferencediameterChord:an interval joining two points onthe circumference chord 7. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to thecircumferenceDiameter: an interval passing through the radius centre, joining any two pointson the circumferencediameterChord:an interval joining two points on secantthe circumferenceSecant: a line that cuts the circle chord 8. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to thecircumferenceDiameter: an interval passing through the radius centre, joining any two pointson the circumferencediameterChord:an interval joining two points on secantthe circumferenceSecant: a line that cuts the circle chordTangent: a line that touches the circletangent 9. Circle GeometryCircle Geometry DefinitionsRadius: an interval joining centre to the arccircumferenceDiameter: an interval passing through the radius centre, joining any two pointson the circumferencediameterChord:an interval joining two points on secantthe circumferenceSecant: a line that cuts the circle chordTangent: a line that touches the circle tangentArc: a piece of the circumference 10. Sector: a plane figure with two radii and an arc as boundaries.sector 11. Sector: a plane figure with two radii and an arc as boundaries. The minor sector is the small piece of the pie, the major sector is the large piecesector 12. Sector: a plane figure with two radii and an arc as boundaries. The minor sector is the small piece of the pie, the major sector is the large pieceA quadrant is a sector where the angle atsectorthe centre is 90 degrees 13. Sector: a plane figure with two radii and an arc as boundaries. The minor sector is the small piece of the pie, the major sector is the large piece A quadrant is a sector where the angle atsector the centre is 90 degreessegment Segment: a plane figure with a chord and anarc as boundaries. 14. Sector: a plane figure with two radii and an arc as boundaries. The minor sector is the small piece of the pie, the major sector is the large piece A quadrant is a sector where the angle atsector the centre is 90 degreessegment Segment: a plane figure with a chord and an arc as boundaries. A semicircle is a segment where the chord is the diameter, it is also a sector as the diameter is two radii. 15. Concyclic Points: points that lie on the same circle. AB D C 16. Concyclic Points: points that lie on the same circle. AB concyclic points D C 17. Concyclic Points: points that lie on the same circle. AB cyclic quadrilateral concyclic points DCCyclic Quadrilateral: a four sided shape with all vertices on thesame circle. 18. AB 19. AB represents the anglesubtended at the centre bythe arc AB 20. AB represents the anglesubtended at the centre bythe arc AB 21. AB represents the anglesubtended at the centre bythe arc AB represents the anglesubtended at thecircumference by the arc AB 22. AB represents the anglesubtended at the centre bythe arc AB represents the anglesubtended at thecircumference by the arc AB 23. AB represents the angleConcentric circles have thesubtended at the centre bysame centre.the arc AB represents the anglesubtended at thecircumference by the arc AB 24. Circles touching internallyshare a common tangent. 25. Circles touching internallyshare a common tangent. Circles touching externally share a common tangent. 26. Circles touching internallyshare a common tangent. Circles touching externally share a common tangent. 27. Circles touching internallyshare a common tangent. Circles touching externally share a common tangent. 28. Chord (Arc) Theorems 29. Chord (Arc) Theorems Note: = chords cut off = arcs 30. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. 31. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. A X B O 32. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA XBO 33. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX XBO 34. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXBO 35. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO 36. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO AXO BXO 90 given R 37. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO AXO BXO 90 given R AO BO radii H 38. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO AXO BXO 90 given R AO BO radii H OX is commonS 39. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO AXO BXO 90 given R AO BO radii H OX is commonS AXO BXO RHS 40. Chord (Arc) Theorems Note: = chords cut off = arcs(1) A perpendicular drawn to a chord from the centre of a circle bisectsthe chord, and the perpendicular bisector of a chord passes throughthe centre. AX BX from centre, bisects chordA Data : AB OX X Prove : AX BXB Proof: Join OA, OBO AXO BXO 90 given R AO BO radii H OX is commonS AXO BXO RHS AX BX matching sides in s 41. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles. 42. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles. A XB O 43. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX AB line joining centre to midpoint, to chord A XB O 44. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX XB O 45. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB O 46. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB 47. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S 48. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S 49. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS 50. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS AXO BXO SSS 51. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching s in s 52. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching s in s AXO BXO 180 straight AXB 53. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching s in s AXO BXO 180 straight AXB 2AXO 180 AXO 90 54. (2) Converse of (1)The line from the centre of a circle to the midpoint of the chord atright angles.OX ABline joining centre to midpoint, to chord AData : AX BX X Prove : AB OXB OProof: Join OA, OB AX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching s in s AXO BXO 180 straight AXB 2AXO 180 AXO 90 AB OX 55. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre. 56. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre. A O XBCY D 57. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre A O XBCY D 58. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centre A O XBCY D 59. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA O XBCY D 60. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O XBCY D 61. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC X BCYD 62. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given B C Y D 63. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 64. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 65. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 AX CY S 66. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 AX CYS AXO CYO 90given R 67. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 AX CYS AXO CYO 90given R OA OC radii H 68. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 AX CYS AXO CYO 90given R OA OC radii H AXO CYO RHS 69. (3) Equal chords of a circle are the same distance from the centre andsubtend equal angles at the centre.OX OY chords, equidistant from centre AOB COD chords subtend s at centreData: AB CD, OX AB, OY CDA Prove : OX OY O Proof: Join OA, OC XAB CD given 1 C Y D BAX AB bisects chord 2 1CY CD bisects chord 2 AX CYS AXO CYO 90given R OA OC radii H AXO CYO RHS OX OY matching sides in s 70. AO BC D 71. Data : AB CD AO BC D 72. Data : AB CD A Prove : AOB CODO BC D 73. Data : AB CD A Prove : AOB CODO Proof: AB CDgiven BC D 74. Data : AB CD A Prove : AOB CODO Proof: AB CDgiven AO BO CO DO radii BC D 75. Data : AB CD A Prove : AOB CODO Proof: AB CDgiven AO BO CO DO radii C D B AOB CODSSS 76. Data : AB CD A Prove : AOB CODO Proof: AB CDgiven AO BO CO DO radii C D B AOB COD SSS AOB COD matching s in s 77. Data : AB CD A Prove : AOB CODO Proof: AB CDgiven AO BO CO DO radii CDB AOB CODSSS AOB COD matching s in s Exercise 9A; 1 ce, 2 aceg, 3, 4, 9, 10 ac, 11 ac, 13, 15, 16, 18