1. circles and theorems - introduction

circles and theorems

1. What are some theorems on tangent of circles? Learning Objective

2. To revise concepts on circles To formulate theorems on tangents Learning Outcomes

3. Circles Let’s revise! What is a circle?

4. Circles Let’s revise! What is point O? The collection of all points on a plane , which are at a fixed distance from a fixed point on the plane is called a circle . O

5. Circles Let’s revise! What is segment OR? The collection of all points on a plane , which are at a fixed distance from a fixed point on the plane is called a circle . The fixed point (O) is called the centre of the circle . O R

6. Circles Let’s revise! The collection of all points on a plane , which are at a fixed distance from a fixed point on the plane is called a circle . The fixed point (O) is called the centre of the circle . The fixed distance (OR) from the centre to the circle is called the radius of the circle. O R

7. Circles Let’s revise! The circle divides a plane into three parts. O

8. Circles Let’s revise! What do you notice? The circle divides a plane into three parts: (i) interior of the circle O

9. Circles Let’s revise! What do you notice? The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle O

10. Circles Let’s revise! The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle (iii) circle itself O

11. Circles Let’s revise! The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle (iii) circle itself The circle and its interior is called the ________ region. O

12. Circles Let’s revise! The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle (iii) circle itself The circle and its interior is called the circular region. O

13. Circles Let’s revise! The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle (iii) circle itself The circle and its interior is called the circular region. The length of the complete circle is called its ______________ . O

14. Circles Let’s revise! The circle divides a plane into three parts: (i) interior of the circle (ii) exterior of the circle (iii) circle itself The circle and its interior is called the circular region. The length of the complete circle is called its circumference . O

15. Circles Let’s revise! What is segment AB? O A B

16. Circles Let’s revise! What is segment AB? The segment joining any two points on the circle is called a chord . O A B

17. Circles Let’s revise! What is segment AB now? The segment joining any two points on the circle is called a chord . O A B

18. Circles Let’s revise! What is segment AB now? The segment joining any two points on the circle is called a chord . A chord passing through the centre of the circle is the diameter . O A B

19. Circles Let’s revise! The piece of circle between two points (A and B) is called the ______ . O A B

20. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the _______ arc . O A B

21. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the ______ arc . O A B

22. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the minor arc . O A B

23. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the minor arc . The region between a chord and either of its arcs is called the ____________ . O A B

24. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the minor arc . The region between a chord and either of its arcs is called the segment . The larger region is called the ________ segment . O A B

25. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the minor arc . The region between a chord and either of its arcs is called the segment . The larger region is called the major segment and the smaller region is called the ________ segment . O A B

26. Circles Let’s revise! The piece of circle between two points (A and B) is called the arc . The longer piece is called the major arc . The shorter piece is called the minor arc . The region between a chord and either of its arcs is called the segment . The larger region is called the major segment and the smaller region is called the minor segment . O A B

27. Circles Let’s revise! What is the slice region called? O M N

28. Circles Let’s revise! The region between an arc and the two radii , joining the center to the end points of the arc is called a sector . O M N

29. Circles Let’s revise! The region between an arc and the two radii , joining the center to the end points of the arc is called a sector . The larger region is called _________ sector . O M N

30. Circles Let’s revise! The region between an arc and the two radii , joining the center to the end points of the arc is called a sector . The larger region is called major sector and the smaller region is called _______ sector . O M N

31. Circles The region between an arc and the two radii , joining the center to the end points of the arc is called a sector . The larger region is called major sector and the smaller region is called minor sector . O M N Let’s revise!

32. Circles Let’s revise! O What is the line called? m

33. Circles Let’s revise! A line touching the circle at a point is called the tangent . O m

34. Circles Let’s revise! A line touching the circle at a point is called the tangent . There can be only one tangent at one point of a circle. O m

35. Circles Let’s revise! A line touching the circle at a point is called the tangent . There can be only one tangent at one point of a circle. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide . O m

36. Circles O M N Well done!

37. To revise concepts on circles To formulate theorems on tangents Learning Outcomes How confident do you feel?

38. To revise concepts on circles To formulate theorems on tangents Learning Outcomes How confident do you feel?

39. Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. Let’s prove it!

40. Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. Proof : We have a circle with centre O and tangent XY at the point P on the circle.

41. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY. Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

42. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

43. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

44. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Why? )   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

45. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Else, XY would be secant )   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

46. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Else, XY would be secant ) OQ>OP   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

47. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Else, XY would be secant ) OQ>OP This happens for every point on line XY, except P.   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

48. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Else, XY would be secant ) OQ>OP This happens for every point on line XY, except P. OP is the shortest distance.   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

49. Proof : We have a circle with centre O and tangent XY at the point P on the circle. Take a point Q ( other than P ) on XY and join OQ. Point Q must lie outside the circle. ( Else, XY would be secant ) OQ>OP This happens for every point on line XY, except P. OP is the shortest distance. OP XY.   Circles Theorem 1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. Hence, proved!

50. Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal. Let’s prove it!

51. Proof : We have a circle with centre O and point P outside the circle. Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

52. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

53. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

54. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Why? )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

55. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

56. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP,   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

57. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

58. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR OP = OP   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

59. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR OP = OP OQP ORP ( Why? )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

60. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR OP = OP OQP ORP ( RHS test )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

61. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR OP = OP OQP ORP ( RHS test ) PQ = PR ( Why? )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal.

62. Proof : We have a circle with centre O and point P outside the circle. We join OP, OQ and OR. ∠OQP = ∠ORP = 90 ( Theorem 1 ) Also, in OQP and ORP, OQ = OR OP = OP OQP ORP ( RHS test ) PQ = PR ( CPCT )   Circles Theorem 2. The lengths of tangents drawn from an external point to a circle are equal. Hence, proved!

63. To revise concepts on circles To formulate theorems on tangents Learning Outcomes How confident do you feel?

64. To revise concepts on circles To formulate theorems on tangents Learning Outcomes How confident do you feel?

65. What are some theorems on tangent of circles? Learning Objective

66. Circles Learning Activity Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.