# 11X1 T13 01 definitions & chord theorems (2010)

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• 1. Circle Geometry

2. Circle Geometry Circle Geometry Definitions 3. Circle Geometry Circle Geometry Definitions 4. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference radius 5. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference Diameter: an interval passing through theradius centre, joining any two points on the circumference diameter 6. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference Diameter: an interval passing through theradius centre, joining any two points on the circumference diameterChord:an interval joining two points on the circumference chord 7. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference Diameter: an interval passing through theradius centre, joining any two points on the circumference diameterChord:an interval joining two points onsecantthe circumference Secant: a line that cuts the circlechord 8. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference Diameter: an interval passing through theradius centre, joining any two points on the circumference diameterChord:an interval joining two points onsecantthe circumference Secant: a line that cuts the circlechordTangent: a line that touches the circle tangent 9. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to thearc circumference Diameter: an interval passing through theradius centre, joining any two points on the circumference diameterChord:an interval joining two points onsecantthe circumference Secant: a line that cuts the circlechordTangent: a line that touches the circletangentArc: a piece of the circumference 10. Sector: a plane figure with two radii and anarc as boundaries.sector 11. Sector: a plane figure with two radii and anarc as boundaries.The minor sector is the small piece of thepie, the major sector is the large piecesector 12. Sector: a plane figure with two radii and anarc as boundaries.The minor sector is the small piece of thepie, the major sector is the large piece A quadrant is a sector where the angle at sectorthe centre is 90 degrees 13. Sector: a plane figure with two radii and anarc as boundaries.The minor sector is the small piece of thepie, the major sector is the large pieceA quadrant is a sector where the angle at sector the centre is 90 degrees segment Segment: a plane figure with a chord and an arc as boundaries. 14. Sector: a plane figure with two radii and anarc as boundaries.The minor sector is the small piece of thepie, the major sector is the large pieceA quadrant is a sector where the angle at sector the centre is 90 degrees segment Segment: a plane figure with a chord and anarc as boundaries.A semicircle is a segment where the chord isthe diameter, it is also a sector as the diameteris two radii. 15. Concyclic Points: points that lie on the same circle. A BD C 16. Concyclic Points: points that lie on the same circle. A B concyclic pointsD C 17. Concyclic Points: points that lie on the same circle. A B cyclic quadrilateral concyclic pointsDC Cyclic Quadrilateral: a four sided shape with all vertices on the same circle. 18. A B 19. A B represents the angle subtended at the centre by the arc AB 20. A B represents the angle subtended at the centre by the arc AB 21. A B represents the angle subtended at the centre by the arc AB represents the angle subtended at the circumference by the arc AB 22. A B represents the angle subtended at the centre by the arc AB represents the angle subtended at the circumference by the arc AB 23. A B represents the angleConcentric circles have the subtended at the centre bysame centre. the arc AB represents the angle subtended at the circumference by the arc AB 24. Circles touching internally share a common tangent. 25. Circles touching internally share a common tangent.Circles touching externallyshare a common tangent. 26. Circles touching internally share a common tangent.Circles touching externallyshare a common tangent. 27. Circles touching internally share a common tangent.Circles touching externallyshare a common tangent. 28. Chord (Arc) Theorems 29. Chord (Arc) TheoremsNote: = chords cut off = arcs 30. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre. 31. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre. AXBO 32. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord AX B O 33. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX B O 34. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B O 35. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB O 36. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB OAXO BXO 90 given R 37. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB OAXO BXO 90 given R AO BO radii H 38. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB OAXO BXO 90 given R AO BO radii H OX is commonS 39. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB OAXO BXO 90 given R AO BO radii H OX is commonS AXO BXO RHS 40. Chord (Arc) TheoremsNote: = chords cut off = arcs (1) A perpendicular drawn to a chord from the centre of a circle bisects the chord, and the perpendicular bisector of a chord passes through the centre.AX BX from centre, bisects chord A Data : AB OXX Prove : AX BX B Proof: Join OA, OB OAXO 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 at right angles. 42. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. AX BO 43. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX AB line joining centre to midpoint, to chordAX BO 44. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXX BO 45. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BO 46. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OB 47. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX BXgiven S 48. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX 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 at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX 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 at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX 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 at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX 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 at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX 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 at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching 's in 's AXO BXO 180 straight AXB 2AXO 180AXO 90 54. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. OX ABline joining centre to midpoint, to chordAData : AX BXXProve : AB OX BOProof: Join OA, OBAX BXgiven S AO BO radii S OX is commonS AXO BXO SSS AXO BXO matching 's in 's AXO BXO 180 straight AXB 2AXO 180AXO 90 AB OX 55. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. 56. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A O X B CY D 57. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centre A O X B CY D 58. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centreA O X B CY D 59. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD A O X B CY D 60. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO X B CY D 61. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCXB CYD 62. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given BC Y D 63. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord2 64. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 65. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 AX CY S 66. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 AX CYS AXO CYO 90given R 67. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 AX CYS AXO CYO 90given R OA OC radii H 68. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 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 and subtend equal angles at the centre. OX OY chords, equidistant from centreAOB COD chords subtend ' s at centre Data: AB CD, OX AB, OY CD AProve : OX OYO Proof: Join OA, OCX AB CD given 1C Y DBAX AB bisects chord21 CY CD bisects chord2 AX CYS AXO CYO 90given R OA OC radii H AXO CYO RHS OX OY matching sides in 's 70. AO B C D 71. Data : AB CDAO B C D 72. Data : AB CDA Prove : AOB CODO B C D 73. Data : AB CDA Prove : AOB CODOProof: AB CDgiven B C D 74. Data : AB CDA Prove : AOB CODOProof: AB CDgiven AO BO CO DO radii B C D 75. Data : AB CDA Prove : AOB CODOProof: AB CDgiven AO BO CO DO radii C DB AOB CODSSS 76. Data : AB CDA Prove : AOB CODOProof: AB CDgiven AO BO CO DO radii C DB AOB COD SSS AOB COD matching 's in 's 77. Data : AB CDA Prove : AOB CODOProof: AB CDgiven AO BO CO DO radii CD B 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