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.