CIRCLE THEOREMS

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

BO O O

A

B

A

TANGENT PROPERTY 1

O

The angle between a tangent and a radius is a right angle.

A

TANGENT PROPERTY 2

O

The two tangents drawn from a point P outside a circle are equal in length.

AP = BPA

P

B

ΙΙ

ΙΙ

O

A

BP

6 cm8 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

tan x

8

6

1 8tan

6x

53.13oAOP

c Shaded area = area of ΔOAP – area of sector OAB

a b

x

21 53.138 6 6

2 360

24 16.69

7.31 cm2 (3 s.f.)

Example

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.

SYMMETRY PROPERTIES OF CHORDS 1

O

A B

The perpendicular line from the centre of a circle to a chord bisects the chord.

ΙΙΙΙ Note: Triangle AOB is isosceles.

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.

ΙΙΙΙ If AB = CD then OP = OQ.

C

D

ΙΙ

ΙΙ

P

Q

ΙΙ

AB = CD

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

BSo angle OBA = x.

THEOREM 1

O

2x

x

The angle at the centre is twice the angle at the circumference.

O

96o

x

Find the value of x.

96 2x

x 96 2

x 48o

Angle at centre = 2 × angle at circumference

Example

O

62o

x

Find the value of x.

x 2 62

x 124o

Angle at centre = 2 × angle at circumference

Example

O

84o

x

Find the value of x.

84 2x

x 84 2

x 42o

Angle at centre = 2 × angle at circumference

Example

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

THEOREM 2

O

An angle in a semi-circle is always a right angle.

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

THEOREM 3

y

x

Opposite angles of a cyclic quadrilateral add up to 180o.

x y 180o

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

THEOREM 4

x

Angles from the same arc in the same segment are equal.

x

x

39o

x

Find the value of x.

x 39o

Angles from the same arc in the same segment are equal.

Example