Circle Circle PropertiesProperties

Part I

A circle is a set of all points in a plane that are the same distance from a

fixed point in a plane

The set of points form the .Circumference

The line joining the centre of a circle and a point on the circumference is called

the……………….Radius

A is a straight line segment joining two points on the circle

chord

A chord that passes through the centre is a

……………………….

diameter

A……………………… is a straight line that cuts the circle in two points

secant

An arc is part of the circumference of a circle

Major arc

Minor arc

A ……………………is part of the circle bounded by two radii and an arc

sector

Minor sector

major sector

A ……………………is part of the circle bounded by a chord and an arc

segment

Minor segment

major segment

The arc AB subtends an angle of at the centre of the circle.

AB

O

Subtends means “to extend under” or “ to be opposite to”

Instructions:

• Draw a circle

• Draw two chords of equal length

• Measure angles AOB and DOC

A

B

C

D

O

What do you notice?

Equal chords subtend equal angles at the centre

Conversely

Equal angles at the centre of a circle stand on equal arcs

Instructions:

• select an arc AB

• subtend the arc AB to the centre O

• subtend the arc AB to a point C on the circumference

• Measure angles AOB and ACB

B

O

A

C

What do you notice?

Instructions:

• select an arc AB

• subtend the arc AB to the centre O

• subtend the arc AB to a point C on the circumference

• Measure angles AOB and ACB

B

O

A

C

What do you notice?

2

The angle that an arc of a circle subtends at the centre is twice the

angle it subtends at the circumference

Instructions:

• select an arc AB

• select two points C, D on the circumference

• subtend the arc AB to a point C on the circumference

• subtend the arc AB to a point D on the circumference

• Measure angles ACB and ADB

B

O

A

C

D

Instructions:

• select an arc AB

• select two points C, D on the circumference

• subtend the arc AB to a point C on the circumference

• subtend the arc AB to a point D on the circumference

• Measure angles ACB and ADB

B

O

A

C

D

What do you notice?

Angles subtended at the circumference by the same arc are equal

Instructions:

• Draw a circle and its diameter

• subtend the diameter to a point on the circumference

•Measure ACB C

B

What do you notice?

A

An angle in a semicircle is a right

angle

γ

Instructions:

•Draw a cyclic quadrilateral (the vertices of the quadrilateral lie on the circumference

•Measure all four angles

β

What do you notice?

180-

The opposite angles of a cyclic quadrilateral are supplementary

180-

180-

If the opposite angles of a quadrilateral are supplementary the quadrilateral is cyclic

β

Instructions:

• Draw a cyclic quadrilateral

• Produce a side of the quadrilateral

•Measure angles and β

If a side of a cyclic quadrilateral is produced, the exterior angle is equal to

the interior opposite angle

Circle Circle PropertiesProperties

Part II tangent properties

A tangent to a circle is a straight line that touches A tangent to a circle is a straight line that touches the circle in one point onlythe circle in one point only

Tangent to a circleTangent to a circleis perpendicular to is perpendicular to the the radius drawn from the point of contact.radius drawn from the point of contact.

Tangents to a circleTangents to a circlefrom an exterior from an exterior pointpoint

are equalare equal

When two circles touch,When two circles touch,the line through their the line through their centrescentres

passes through their point of contactpasses through their point of contact

Point of contact

External Contact

When two circles touch,When two circles touch,the line through their the line through their centrescentres

passes through their point of contactpasses through their point of contact

Point of contact

Internal Contact

The angle between a The angle between a tangent tangent and a chord through the point of and a chord through the point of

contact contact is equal to the angle in the alternate is equal to the angle in the alternate segmentsegment

The square of the length of the tangentThe square of the length of the tangent

from an external point is equal tofrom an external point is equal to

the product of the intercepts of the secantthe product of the intercepts of the secant

passing through this pointpassing through this point

AA

BB

BA2=BC.BD

CC

DD

B=external point

The square of the length of the tangentThe square of the length of the tangent

from an external point is equal tofrom an external point is equal to

the product of the intercepts of the secantthe product of the intercepts of the secant

passing through this pointpassing through this point

AA

BB

BA2=BC.BD

CC

DD

Note: B is the crucial point in the formula

Circle Circle PropertiesProperties

Chord properties

A

B

C

D

X

AX.XB=CX.XD

Triangle AXD is similar to triangle CXB hence

A

B

C

D

X

AX.XB=CX.XD

Note: X is the crucial point in the formula

Chord AB and CD intersect at X

Prove AX.XB=CX.XD

A

B

C

D

X

In AXD and CXB

AXD = CXB (Vertically Opposite Angles)

DAX = BCX (Angles standing on same arc)

ADX = CBX (Angles standing on same arc)

AXD CXB

Hence (Equiangular )XB

CX

XD

AX

XDCXXBAX .. AAA test for similar triangles

A

B

C

A perpendicular line from the centre off a circle A perpendicular line from the centre off a circle to a chord bisects the chord to a chord bisects the chord

A

B

C

Conversley: A line from the centre of a circle Conversley: A line from the centre of a circle that bisects a chord is perpendicular to the that bisects a chord is perpendicular to the chord chord

A

B

C

Equal chords are equidistant from the centre of Equal chords are equidistant from the centre of the circle the circle

A

B

C

Conversley: Chords that are equidistant from the Conversley: Chords that are equidistant from the centre are equalcentre are equal

Quick Quick QuizQuiz

a

40

a= 40

b

40

b= 80C

d

60 d= 120C

f

55 f= 55C

m=62

C62

m

e

e= 90C

x= 12

C102

102

12 cm

x cm

k

70

k= 35C

a

120

a= 50

10

x100

x= 50C

y y= 55C

35

Quick Quick QuizQuiz

answer=A

10575

Which quadrilateral is concyclic?

A

B

C

100

110

20

140

c

60 c =60C

Tangent

g

g= 90C

Tangent

h= 4C

4cm

h cm

Tangent

Tangent

m

40

m =50C

Tangent

y =50

y

a= 65C

50

Q

a

P

R

PQ, RQ are tangents

n= 5

C

10

4

8

n

nx8=4x10

8n =40

n =5

q= 25

C

10

4

q4q=10

2

4q=100

q=25

x= 12

C

8

4

x

4(4+x)=82

4(4+x)=64

4+x=16

x=12

BA2=BC.BD

k= 5

C

8m

k

3m

K2=32+42

K =5