11X1 T07 01 definitions & chord theorems

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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 radius Diameter: an interval passing through the centre, joining any two points on the circumference diameter
  • 6. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference radius Diameter: an interval passing through the centre, joining any two points on the circumference diameter Chord: an interval joining two points on the circumference chord
  • 7. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference radius Diameter: an interval passing through the centre, joining any two points on the circumference diameter Chord: an interval joining two points on the circumference chord Secant: a line that cuts the circle secant
  • 8. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference radius Diameter: an interval passing through the centre, joining any two points on the circumference diameter Chord: an interval joining two points on the circumference chord Secant: a line that cuts the circle secant Tangent: a line that touches the circle tangent
  • 9. Circle Geometry Circle Geometry Definitions Radius: an interval joining centre to the circumference radius Diameter: an interval passing through the centre, joining any two points on the circumference diameter Chord: an interval joining two points on the circumference chord Secant: a line that cuts the circle secant Tangent: a line that touches the circle tangent Arc: a piece of the circumference arc
  • 10.  
  • 11. Sector: a plane figure with two radii and an arc as boundaries. sector
  • 12. Sector: a plane figure with two radii and an arc as boundaries. sector The minor sector is the small “piece of the pie”, the major sector is the large “piece”
  • 13. Sector: a plane figure with two radii and an arc as boundaries. sector 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 at the centre is 90 degrees
  • 14. Sector: a plane figure with two radii and an arc as boundaries. sector 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 at the centre is 90 degrees Segment: a plane figure with a chord and an arc as boundaries. segment
  • 15. Sector: a plane figure with two radii and an arc as boundaries. sector 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 at the centre is 90 degrees Segment: a plane figure with a chord and an arc as boundaries. segment A semicircle is a segment where the chord is the diameter, it is also a sector as the diameter is two radii.
  • 16.  
  • 17. Concyclic Points: points that lie on the same circle. A B C D
  • 18. Concyclic Points: points that lie on the same circle. concyclic points A B C D
  • 19. Concyclic Points: points that lie on the same circle. concyclic points Cyclic Quadrilateral: a four sided shape with all vertices on the same circle. cyclic quadrilateral A B C D
  • 20.  
  • 21. A B
  • 22. A B represents the angle subtended at the centre by the arc AB
  • 23. A B represents the angle subtended at the centre by the arc AB
  • 24. A B represents the angle subtended at the centre by the arc AB represents the angle subtended at the circumference by the arc AB
  • 25. A B represents the angle subtended at the centre by the arc AB represents the angle subtended at the circumference by the arc AB
  • 26. A B represents the angle subtended at the centre by the arc AB represents the angle subtended at the circumference by the arc AB Concentric circles have the same centre.
  • 27.  
  • 28.  
  • 29. Circles touching internally share a common tangent.
  • 30. Circles touching internally share a common tangent. Circles touching externally share a common tangent.
  • 31. Circles touching internally share a common tangent. Circles touching externally share a common tangent.
  • 32. Circles touching internally share a common tangent. Circles touching externally share a common tangent.
  • 33. Chord (Arc) Theorems
  • 34. Chord (Arc) Theorems Note: = chords cut off = arcs
  • 35. Chord (Arc) Theorems Note: = 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.
  • 36. Chord (Arc) Theorems Note: = 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. A B O X
  • 37. Chord (Arc) Theorems Note: = 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. A B O X
  • 38. Chord (Arc) Theorems Note: = 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. A B O X
  • 39. Chord (Arc) Theorems Note: = 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. A B O X
  • 40. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 41. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 42. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 43. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 44. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 45. Chord (Arc) Theorems Note: = 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. Proof: Join OA , OB A B O X
  • 46. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles.
  • 47. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. A B O X
  • 48. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. A B O X
  • 49. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. A B O X
  • 50. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. A B O X
  • 51. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 52. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 53. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 54. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 55. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 56. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 57. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 58. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 59. (2) Converse of (1) The line from the centre of a circle to the midpoint of the chord at right angles. Proof: Join OA , OB A B O X
  • 60. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre.
  • 61. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A B O X C D Y
  • 62. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A B O X C D Y
  • 63. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A B O X C D Y
  • 64. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A B O X C D Y
  • 65. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. A B O X C D Y
  • 66. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 67. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 68. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 69. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 70. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 71. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 72. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 73. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 74. (3) Equal chords of a circle are the same distance from the centre and subtend equal angles at the centre. Proof: Join OA , OC A B O X C D Y
  • 75. A B O C D
  • 76. A B O C D
  • 77. A B O C D
  • 78. Proof: A B O C D
  • 79. Proof: A B O C D
  • 80. Proof: A B O C D
  • 81. Proof: A B O C D
  • 82. Proof: Exercise 9A; 1 ce, 2 aceg, 3, 4, 9, 10 ac, 11 ac, 13, 15, 16, 18 A B O C D