© T Madas O O O O O O O The Circle Theorems. © T Madas 1 st Theorem.

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OO

O

O

O

OO

The Circle Theorems

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1st Theorem

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The perpendicular bisector of a chord passes through the centre of the circle

O

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O

The perpendicular bisector of a chord passes through the centre of the circle

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O

The perpendicular bisector of a chord passes through the centre of the circle

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Finding the Centre of Rotation

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The shapes below have been produced by rotation.Find the centre of rotation

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The shapes below have been produced by rotation.Find the centre of rotation

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The shapes below have been produced by rotation.Find the centre of rotation

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The shapes below have been produced by rotation.Find the centre of rotation

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2nd Theorem

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O

Inscribed angles which correspond to the same arc are equal

Inscribed Angle

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O

Inscribed angles which correspond to the same arc are equal

Does this inscribed angle correspond to the same arc?

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3rd Theorem

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A central angle is twice as large as any inscribed angle which corresponds to the same arc

Central Angle

Inscribed Angle

O

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Various Forms of the Theorem

O

O

O

OO

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4th Theorem

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O

An inscribed angle which corresponds to a diameter (or semicircle) is a right angle

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5th Theorem

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O Cyclic Quadrilateral

Opposite angles in a cyclic quadrilateral are supplementary

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6th Theorem

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O

Tangent

Tangent point

A tangent and a radius drawn at any point on the circumference of the circle meet at right angles

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7th T

heore

m

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O

The intersection of two tangents to a circle is equidistant from their points of contact.

[Their angle of intersection and the central angle formed by the radii at the points of contact, are supplementary]

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8th Theorem

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O

segment

segment

sector

segmen

t

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O

AlternatingSegments

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O

The angle formed by a chord and a tangent at one of its endpoints is equal to the inscribed angle corresponding to the same chord in the alternating segment

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Circle Theorem Test

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Circle Theorem Mini Test

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Practice Question

1

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O

30°

x

45°30°

15°150°

15°

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Practice Question

2

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50°

z100°

50°

30°

x

y

30°

O

© T Madas

Practice Question

3

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70°

a

b

c

20°

70°

20°

O

© T Madas

Practice Question

4

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95°

n

m55° 85°

40°

40°

p55°

O

© T Madas

Practice Question

5

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25°x

y

25°

Tangent point

65°

O

© T Madas

Practice Question

6

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55°s55°

t110°

O

© T Madas

Practice Question

7

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u

28°

v

28°

56°

O

© T Madas

Practice Question

8

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300°

h

O60°

30°

150°

© T Madas

Practice Question

9

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130°

c

50°

100°

O

© T Madas

Practice Question

10

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50°

a

b

25°

25°

O

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50°

a

b

130°

25°

25°

Can you solve this problem without a circle theorem?

O

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Practice Question

11

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65°

x230°

115°

O

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Practice Question

12

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100°

z

100°

200°

O

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Practice Question

13

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84°

a

b

O

42°

138°

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Practice Question

14

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32

°

gO f148°

32°32°

64°296°

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Practice Question

15

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115°p O

q

65°

90°90°

25°

© T Madas

Practice Question

16

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90°

x

O45° 45°

© T Madas

Practice Question

17

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70°

pOA

B

C

AB = BC

qr

55°

90°

35°

20°

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Practice Question

18

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72°

u

O

v

90°

18°

72°

© T Madas

Practice Question

19

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30°

a

O

b

cTangent point

Tangent point

60°

60°

120°

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Practice Question

20

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O58°

z yx

58°32°58°

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Practice Question

21

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85°

x

O

95°

85°

85°

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Practice Question

22

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57°

t

O

r123°

57°

Can you think of another reason as to why both these angles are 57° ?

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Practice Question

23

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56°

62°62° w

Ox

y

z

124°

56°

62°

118°

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Practice Question

24

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u45°

160°

155°

O25°

20°25°135°

v20°

© T Madas

Practice Question

25

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30°

x

O30°

120°

240°

© T Madas

Practice Question

26

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75° O

x

75°

30°

30°

60°

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Practice Question

27

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72°

x

O144°

18°18°

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Practice Question

28

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40°a

O

b40° 140°

50°

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Practice Question

29

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30°

θ

O60°

60°

30°

© T Madas

Practice Question

30

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5°

n

O

25°

65°

© T Madas

Practice Question

31

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Oa22° b

c d

Tangent point

Tangent point

22°

68°56°124°

68°

68°

Exam question

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