Page 1 © T Madas
OO
O
O
O
OO
The Circle Theorems
Page 2 © T Madas
1st Theorem
Page 3 © T Madas
The perpendicular bisector of a chord passes through the centre of the circle
O
Page 4 © T Madas
O
The perpendicular bisector of a chord passes through the centre of the circle
Page 5 © T Madas
O
The perpendicular bisector of a chord passes through the centre of the circle
Page 6 © T Madas
Finding the Centre of Rotation
Page 7 © T Madas
The shapes below have been produced by rotation.Find the centre of rotation
Page 8 © T Madas
The shapes below have been produced by rotation.Find the centre of rotation
Page 9 © T Madas
The shapes below have been produced by rotation.Find the centre of rotation
Page 10 © T Madas
The shapes below have been produced by rotation.Find the centre of rotation
Page 11 © T Madas
2nd Theorem
Page 12 © T Madas
O
Inscribed angles which correspond to the same arc are equal
Inscribed Angle
Page 13 © T Madas
O
Inscribed angles which correspond to the same arc are equal
Does this inscribed angle correspond to the same arc?
Page 14 © T Madas
3rd Theorem
Page 15 © T Madas
A central angle is twice as large as any inscribed angle which corresponds to the same arc
Central Angle
Inscribed Angle
O
Page 16 © T Madas
Various Forms of the Theorem
O
O
O
OO
Page 17 © T Madas
4th Theorem
Page 18 © T Madas
O
An inscribed angle which corresponds to a diameter (or semicircle) is a right angle
Page 19 © T Madas
5th Theorem
Page 20 © T Madas
O Cyclic Quadrilateral
Opposite angles in a cyclic quadrilateral are supplementary
Page 21 © T Madas
6th Theorem
Page 22 © 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
Page 23 © T Madas
7th T
heore
m
Page 24 © 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]
Page 25 © T Madas
8th Theorem
Page 26 © T Madas
O
segment
segment
sector
segmen
t
Page 27 © T Madas
O
AlternatingSegments
Page 28 © 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
Page 29 © T Madas
Circle Theorem Test
Page 30 © T Madas
Circle Theorem Mini Test
Page 32 © T Madas
Practice Question
1
Page 33 © T Madas
O
30°
x
45°30°
15°150°
15°
Page 34 © T Madas
Practice Question
2
Page 35 © T Madas
50°
z100°
50°
30°
x
y
30°
O
Page 36 © T Madas
Practice Question
3
Page 37 © T Madas
70°
a
b
c
20°
70°
20°
O
Page 38 © T Madas
Practice Question
4
Page 39 © T Madas
95°
n
m55° 85°
40°
40°
p55°
O
Page 40 © T Madas
Practice Question
5
Page 41 © T Madas
25°x
y
25°
Tangent point
65°
O
Page 42 © T Madas
Practice Question
6
Page 43 © T Madas
55°s55°
t110°
O
Page 44 © T Madas
Practice Question
7
Page 45 © T Madas
u
28°
v
28°
56°
O
Page 46 © T Madas
Practice Question
8
Page 47 © T Madas
300°
h
O60°
30°
150°
Page 48 © T Madas
Practice Question
9
Page 49 © T Madas
130°
c
50°
100°
O
Page 50 © T Madas
Practice Question
10
Page 51 © T Madas
50°
a
b
25°
25°
O
Page 52 © T Madas
50°
a
b
130°
25°
25°
Can you solve this problem without a circle theorem?
O
Page 53 © T Madas
Practice Question
11
Page 54 © T Madas
65°
x230°
115°
O
Page 55 © T Madas
Practice Question
12
Page 56 © T Madas
100°
z
100°
200°
O
Page 57 © T Madas
Practice Question
13
Page 58 © T Madas
84°
a
b
O
42°
138°
Page 59 © T Madas
Practice Question
14
Page 60 © T Madas
32
°
gO f148°
32°32°
64°296°
Page 61 © T Madas
Practice Question
15
Page 62 © T Madas
115°p O
q
65°
90°90°
25°
Page 63 © T Madas
Practice Question
16
Page 64 © T Madas
90°
x
O45° 45°
Page 65 © T Madas
Practice Question
17
Page 66 © T Madas
70°
pOA
B
C
AB = BC
qr
55°
90°
35°
20°
Page 67 © T Madas
Practice Question
18
Page 68 © T Madas
72°
u
O
v
90°
18°
72°
Page 69 © T Madas
Practice Question
19
Page 70 © T Madas
30°
a
O
b
cTangent point
Tangent point
60°
60°
120°
Page 71 © T Madas
Practice Question
20
Page 72 © T Madas
O58°
z yx
58°32°58°
Page 73 © T Madas
Practice Question
21
Page 74 © T Madas
85°
x
O
95°
85°
85°
Page 75 © T Madas
Practice Question
22
Page 76 © T Madas
57°
t
O
r123°
57°
Can you think of another reason as to why both these angles are 57° ?
Page 77 © T Madas
Practice Question
23
Page 78 © T Madas
56°
62°62° w
Ox
y
z
124°
56°
62°
118°
Page 79 © T Madas
Practice Question
24
Page 80 © T Madas
u45°
160°
155°
O25°
20°25°135°
v20°
Page 81 © T Madas
Practice Question
25
Page 82 © T Madas
30°
x
O30°
120°
240°
Page 83 © T Madas
Practice Question
26
Page 84 © T Madas
75° O
x
75°
30°
30°
60°
Page 85 © T Madas
Practice Question
27
Page 86 © T Madas
72°
x
O144°
18°18°
Page 87 © T Madas
Practice Question
28
Page 88 © T Madas
40°a
O
b40° 140°
50°
Page 89 © T Madas
Practice Question
29
Page 90 © T Madas
30°
θ
O60°
60°
30°
Page 91 © T Madas
Practice Question
30
Page 92 © T Madas2
5°
n
O
25°
65°
Page 93 © T Madas
Practice Question
31
Page 94 © T Madas
Oa22° b
c d
Tangent point
Tangent point
22°
68°56°124°
68°
68°
Exam question