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Circle Properties

Circle Properties. Part I . A circle is a set of all points in a plane that are the same distance from a fixed point in a plane. Circumference. The set of points form the. The line joining the centre of a circle and a point on the circumference is called the………………. Radius. chord.

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Circle Properties

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  1. Circle Properties Part I

  2. A circle is a set of all points in a plane that are the same distance from a fixed point in a plane Circumference The set of points form the .

  3. The line joining the centre of a circle and a point on the circumference is called the………………. Radius

  4. chord Ais a straight line segment joining two points on the circle

  5. A chord that passes through the centre is a ………………………. diameter

  6. secant A……………………… is a straight line that cuts the circle in two points

  7. An arc is part of the circumference of a circle Major arc Minor arc

  8. sector A ……………………is part of the circle bounded by two radii and an arc major sector Minor sector

  9. segment A ……………………is part of the circle bounded by a chord and an arc major segment Minor segment

  10. The arc AB subtends an angle of at the centre of the circle. Subtends means “to extend under” or “ to be opposite to” O  B A

  11. Instructions: • Draw a circle • Draw two chords of equal length • Measure angles AOB and DOC B O A C D What do you notice?

  12. Equal chords subtend equal angles at the centre  

  13. Conversely Equal angles at the centre of a circle stand on equal arcs  

  14. Instructions: • select an arc AB • subtend the arc AB to the centre O • subtend the arc AB to a point C on the circumference • Measure angles  AOB and  ACB C O A What do you notice? B

  15. Instructions: • select an arc AB • subtend the arc AB to the centre O • subtend the arc AB to a point C on the circumference • Measure angles  AOB and  ACB C O A What do you notice? B

  16. The angle that an arc of a circle subtends at the centre is twice the angle it subtends at the circumference  2

  17. Instructions: • select an arc AB • select two points C, D on the circumference • subtend the arc AB to a point C on the circumference • subtend the arc AB to a point D on the circumference • Measure angles  ACB and  ADB D C O A B

  18. Instructions: • select an arc AB • select two points C, D on the circumference • subtend the arc AB to a point C on the circumference • subtend the arc AB to a point D on the circumference • Measure angles  ACB and  ADB D C O A What do you notice? B

  19.  Angles subtended at the circumference by the same arc are equal  

  20. Instructions: • Draw a circle and its diameter • subtend the diameter to a point on the circumference • Measure ACB C A B What do you notice?

  21. An angle in a semicircle is a right angle

  22. Instructions: • Draw a cyclic quadrilateral (the vertices of the quadrilateral lie on the circumference • Measure all four angles   γ β What do you notice?

  23. The opposite angles of a cyclic quadrilateral are supplementary 180-  180- 

  24. If the opposite angles of a quadrilateral are supplementary the quadrilateral is cyclic 180- 

  25. Instructions: • Draw a cyclic quadrilateral • Produce a side of the quadrilateral • Measure angles  and β β 

  26. If a side of a cyclic quadrilateral is produced, the exterior angle is equal to the interior opposite angle  

  27. Circle Properties Part II tangent properties

  28. A tangent to a circle is a straight line that touches the circle in one point only

  29. Tangent to a circle is perpendicular to the radius drawn from the point of contact.

  30. Tangents to a circle from an exterior point are equal

  31. When two circles touch, the line through their centres passes through their point of contact External Contact Point of contact

  32. When two circles touch, the line through their centres passes through their point of contact Internal Contact Point of contact

  33. The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment  

  34. The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point A B B=external point C D BA2=BC.BD

  35. The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point A Note: B is the crucial point in the formula B C D BA2=BC.BD

  36. Circle Properties Chord properties

  37. Triangle AXD is similar to triangle CXB hence C A X D B AX.XB=CX.XD

  38. Note: X is the crucial point in the formula C A X D B AX.XB=CX.XD

  39. Chord AB and CD intersect at X Prove AX.XB=CX.XD In AXD and CXB (Vertically Opposite Angles) AXD =  CXB C (Angles standing on same arc) DAX =  BCX A X ADX =  CBX (Angles standing on same arc) B D  AXD    CXB Hence (Equiangular ) AAA test for similar triangles

  40. A perpendicular line from the centre off a circle to a chord bisects the chord C A B

  41. Conversley: A line from the centre of a circle that bisects a chord is perpendicular to the chord C A B

  42. Equal chords are equidistant from the centre of the circle C A B

  43. Conversley: Chords that are equidistant from the centre are equal C A B

  44. Quick Quiz

  45. a a= 40 40

  46. 40 b= 80 C b

  47. d d= 120 60 C

  48. f f= 55 55 C

  49. m= 62 62 C m

  50. e e= 90 C

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