This document summarizes several theorems and properties related to circles, tangents, chords, and angles. It states that a tangent line forms a right angle with a radius and that two tangents from an external point are equal in length. Chords are line segments joining two points on a circle, and a chord bisects the perpendicular line from the center. The angle at the center of a circle is twice the angle at the circumference. An angle in a semi-circle is a right angle. Opposite angles of a cyclic quadrilateral sum to 180 degrees, and angles from the same arc in the same segment are equal. Several examples demonstrate applying these circle theorems and properties.

1. CIRCLE THEOREMS

2. TANGENTS
A straight line can intersect a circle in three possible ways.
It can be:
A DIAMETER A CHORD A TANGENT
2 points of
intersection
2 points of
intersection
1 point of
intersection
A
B
O O O
A
B
A

3. TANGENT PROPERTY 1
O
The angle between a
tangent and a radius is a
right angle.
A

4. TANGENT PROPERTY 2
O
The two tangents drawn
from a point P outside a
circle are equal in length.
AP = BP
A
P
B

5. O
A
B
P
6 cm
8 cm
AP is a tangent to the circle.
a Calculate the length of OP.
b Calculate the size of angle AOP.
c Calculate the shaded area.
OP2
 62
 82
OP2
 100
OP  10 cm
tanx 
8
6
  
  
 
1 8
tan
6
x
 53.13o
AOP
c Shaded area = area of ΔOAP – area of sector OAB
a b
x

   
     
   
   
2
1 53.13
8 6 6
2 360
 24 16.69
 7.31cm2
(3 s.f.)
Example

6. CHORDS AND SEGMENTS
major segment
minor segment
A straight line joining two points on the circumference of a
circle is called a chord.
A chord divides a circle into two segments.

7. SYMMETRY PROPERTIES OF CHORDS 1
O
A B
The perpendicular line from the
centre of a circle to a chord bisects
the chord.
ΙΙ
ΙΙ Note: Triangle AOB is isosceles.

8. SYMMETRY PROPERTIES OF CHORDS 2
O
A B
If two chords AB and CD are the
same length then they will be the
same perpendicular distance from
the centre of the circle.
ΙΙ
ΙΙ If AB = CD then OP = OQ.
C
D
P
Q
Ι
AB = CD

9. O
96o
x
Find the value of x.
2x  96  180
2x  84
x  42o
Triangle OAB is isosceles
because OA = OB (radii of circle)
Example
A
B
So angle OBA = x.

10. THEOREM 1
O
2x
x
The angle at the centre is
twice the angle at the
circumference.

11. O
96o
x
Find the value of x.
96  2x
x  96  2
x  48o
Angle at centre
= 2 × angle at circumference
Example

12. O
62o
x
Find the value of x.
x  2  62
x  124o
Angle at centre
= 2 × angle at circumference
Example

13. O
84o
x
Find the value of x.
84  2x
x  84  2
x  42o
Angle at centre
= 2 × angle at circumference
Example

14. O
104o
x
Find the value of x.
y  2 104
y  208
Angle at centre
= 2 × angle at circumference
y
x  360  208
x  152o
Example

15. THEOREM 2
O
An angle in a semi-circle
is always a right angle.

16. O
Find the value of x.
x  58  90  180
x  32o
Angles in a semi-circle = 90o
and
angles in a triangle add up to 180o
.
58o
x
Example

17. THEOREM 3
y
x
Opposite angles of a
cyclic quadrilateral add
up to 180o
.
x  y  180o

18. Find the values of x and y.
x 132  180
x  48o
Opposite angles in a cyclic
quadrilateral add up to 180o
.
x
y
75o 132o
y  75  180
y  105o
Example

19. THEOREM 4
x
Angles from the same
arc in the same segment
are equal.
x
x

20. 39o
x
Find the value of x.
x  39o
Angles from the same arc in the same
segment are equal.
Example