Circle Properties
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Circle Properties. 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. Circumference. The set of points form the. The line joining the centre of a circle and a point on the circumference is called the………………. Radius. chord.

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

Circumference

The set of points form the .


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

the……………….

Radius


chord circumference is called

Ais a straight line segment joining two points on the circle


A chord that passes through the centre is a circumference is called

……………………….

diameter


secant circumference is called

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


An circumference is called arc is part of the circumference of a circle

Major arc

Minor arc


sector circumference is called

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

major sector

Minor sector


segment circumference is called

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

major segment

Minor segment


The arc AB subtends an angle of circumference is called at the centre of the circle.

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

O

B

A


  • Instructions: circumference is called

  • Draw a circle

  • Draw two chords of equal length

  • Measure angles AOB and DOC

B

O

A

C

D

What do you notice?


Equal chords subtend equal angles at the centre circumference is called


Conversely circumference is called

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


  • Instructions: circumference is called

  • 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

C

O

A

What do you notice?

B


  • Instructions: circumference is called

  • 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

C

O

A

What do you notice?

B


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

2


  • Instructions: twice the angle it subtends at the circumference

  • 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

D

C

O

A

B


  • Instructions: twice the angle it subtends at the circumference

  • 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

D

C

O

A

What do you notice?

B


twice the angle it subtends at the circumference

Angles subtended at the circumference by the same arc are equal


  • Instructions: twice the angle it subtends at the circumference

  • Draw a circle and its diameter

  • subtend the diameter to a point on the circumference

  • Measure ACB

C

A

B

What do you notice?


An angle in a semicircle is a right angle twice the angle it subtends at the circumference


  • Instructions: twice the angle it subtends at the circumference

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

  • Measure all four angles

γ

β

What do you notice?


The opposite angles of a cyclic quadrilateral are supplementary

180-

180-


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

180-


  • Instructions: the quadrilateral is cyclic

  • Draw a cyclic quadrilateral

  • Produce a side of the quadrilateral

  • Measure angles  and β

β


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


Circle Properties exterior angle is equal to the interior opposite angle

Part II tangent properties



Tangent to a circle circle in one point only

is perpendicular to the

radius drawn from the point of contact.


Tangents to a circle circle in one point only

from an exterior point

are equal


When two circles touch, circle in one point only

the line through their centres

passes through their point of contact

External Contact

Point of contact


When two circles touch, circle in one point only

the line through their centres

passes through their point of contact

Internal Contact

Point of contact


The angle between a tangent circle in one point only

and a chord through the point of contact

is equal to the angle in the alternate segment


The square of the length of the tangent circle in one point only

from an external point is equal to

the product of the intercepts of the secant

passing through this point

A

B

B=external point

C

D

BA2=BC.BD


The square of the length of the tangent circle in one point only

from an external point is equal to

the product of the intercepts of the secant

passing through this point

A

Note: B is the crucial point in the formula

B

C

D

BA2=BC.BD


Circle Properties circle in one point only

Chord properties


Triangle AXD is similar to triangle CXB hence circle in one point only

C

A

X

D

B

AX.XB=CX.XD


Note: X is the crucial point in the formula circle in one point only

C

A

X

D

B

AX.XB=CX.XD


Chord AB and CD intersect at X circle in one point only

Prove AX.XB=CX.XD

In AXD and CXB

(Vertically Opposite Angles)

AXD =  CXB

C

(Angles standing on same arc)

DAX =  BCX

A

X

ADX =  CBX

(Angles standing on same arc)

B

D

 AXD    CXB

Hence (Equiangular )

AAA test for similar triangles



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

C

A

B


Equal chords are equidistant from the centre of the circle a chord is perpendicular to the chord

C

A

B



Quick Quiz equal


a equal

a=

40

40


40 equal

b=

80

C

b


d equal

d=

120

60

C


f equal

f=

55

55

C


m equal=

62

62

C

m


e equal

e=

90

C


12 cm equal

x=

12

102

C

102

x cm


k equal

k=

35

C

70


a equal

a=

50

120

10


x equal=

50

C

100

x


y equal=

55

y

C

35 


Quick Quiz equal


Which quadrilateral is concyclic? equal

A

105

75

answer=

A

110

B

C

140

100

20


c equal=

60

60

C

Tangent

c


Tangent equal

g

g=

90

C


Tangent equal

4cm

h cm

h=

4

C

Tangent


y equal

m =

50

y =

50

C

Tangent

40

m


Q equal

P

50 

a=

65

a

C

R

PQ, RQ are tangents


10 equal

n=

5

n

C

nx8=4x10

8n =40

n =5

4

8


q equal=

25

q

C

4q=102

4q=100

q=25

4

10


B equalA2=BC.BD

x=

12

4(4+x)=82

4(4+x)=64

4+x=16

x=12

x

C

4

8


k equal=

5

C

3m

K2=32+42

K =5

8m

k


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