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

© T Madas

OO

O

O

O

OO

The Circle Theorems

© T Madas

1st Theorem

© T Madas

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

O

© T Madas

O

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

© T Madas

Finding the Centre of Rotation

© T Madas

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

© T Madas

2nd Theorem

© T Madas

O

Inscribed angles which correspond to the same arc are equal

Inscribed Angle

© T Madas

O

Inscribed angles which correspond to the same arc are equal

Does this inscribed angle correspond to the same arc?

© T Madas

3rd Theorem

© T Madas

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

Central Angle

Inscribed Angle

O

© T Madas

Various Forms of the Theorem

O

O

O

OO

© T Madas

4th Theorem

© T Madas

O

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

© T Madas

5th Theorem

© T Madas

O Cyclic Quadrilateral

Opposite angles in a cyclic quadrilateral are supplementary

© T Madas

6th Theorem

© T Madas

O

Tangent

Tangent point

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

© T Madas

7th T

heore

m

© T Madas

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]

© T Madas

8th Theorem

© T Madas

O

segment

segment

sector

segmen

t

© T Madas

O

AlternatingSegments

© T Madas

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

© T Madas

Circle Theorem Test

© T Madas

Circle Theorem Mini Test

© T Madas

Practice Question

1

© T Madas

O

30°

x

45°30°

15°150°

15°

© T Madas

Practice Question

2

© T Madas

50°

z100°

50°

30°

x

y

30°

O

© T Madas

Practice Question

3

© T Madas

70°

a

b

c

20°

70°

20°

O

© T Madas

Practice Question

4

© T Madas

95°

n

m55° 85°

40°

40°

p55°

O

© T Madas

Practice Question

5

© T Madas

25°x

y

25°

Tangent point

65°

O

© T Madas

Practice Question

6

© T Madas

55°s55°

t110°

O

© T Madas

Practice Question

7

© T Madas

u

28°

v

28°

56°

O

© T Madas

Practice Question

8

© T Madas

300°

h

O60°

30°

150°

© T Madas

Practice Question

9

© T Madas

130°

c

50°

100°

O

© T Madas

Practice Question

10

© T Madas

50°

a

b

25°

25°

O

© T Madas

50°

a

b

130°

25°

25°

Can you solve this problem without a circle theorem?

O

© T Madas

Practice Question

11

© T Madas

65°

x230°

115°

O

© T Madas

Practice Question

12

© T Madas

100°

z

100°

200°

O

© T Madas

Practice Question

13

© T Madas

84°

a

b

O

42°

138°

© T Madas

Practice Question

14

© T Madas

32

°

gO f148°

32°32°

64°296°

© T Madas

Practice Question

15

© T Madas

115°p O

q

65°

90°90°

25°

© T Madas

Practice Question

16

© T Madas

90°

x

O45° 45°

© T Madas

Practice Question

17

© T Madas

70°

pOA

B

C

AB = BC

qr

55°

90°

35°

20°

© T Madas

Practice Question

18

© T Madas

Practice Question

29

© T Madas

30°

θ

O60°

60°

30°

© T Madas

Practice Question

30

© T Madas2

5°

n

O

25°

65°

© T Madas

Practice Question

31

© T Madas

Oa22° b

c d

Tangent point

Tangent point

22°

68°56°124°

68°

68°

Exam question

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

Documents

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