circle properties part i. a circle is a set of all points in a plane that are the same distance from...

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Circle Circle Properties Properties Part I

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Page 1: 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 The set of points form the

Circle Circle PropertiesProperties

Part I

Page 2: 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 The set of points form the

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

Page 3: 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 The set of points form the

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

the……………….Radius

Page 4: 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 The set of points form the

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

chord

Page 5: 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 The set of points form the

A chord that passes through the centre is a

……………………….

diameter

Page 6: 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 The set of points form the

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

secant

Page 7: 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 The set of points form the

An arc is part of the circumference of a circle

Major arc

Minor arc

Page 8: 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 The set of points form the

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

sector

Minor sector

major sector

Page 9: 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 The set of points form the

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

segment

Minor segment

major segment

Page 10: 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 The set of points form the

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”

Page 11: 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 The set of points form the

Instructions:

• Draw a circle

• Draw two chords of equal length

• Measure angles AOB and DOC

A

B

C

D

O

What do you notice?

Page 12: 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 The set of points form the

Equal chords subtend equal angles at the centre

Page 13: 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 The set of points form the

Conversely

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

Page 14: 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 The set of points form the

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?

Page 15: 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 The set of points form the

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?

Page 16: 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 The set of points form the

2

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

angle it subtends at the circumference

Page 17: 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 The set of points form the

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

Page 18: 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 The set of points form the

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?

Page 19: 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 The set of points form the

Angles subtended at the circumference by the same arc are equal

Page 20: 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 The set of points form the

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

Page 21: 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 The set of points form the

An angle in a semicircle is a right

angle

Page 22: 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 The set of points form the

γ

Instructions:

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

•Measure all four angles

β

What do you notice?

Page 23: 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 The set of points form the

180-

The opposite angles of a cyclic quadrilateral are supplementary

180-

Page 24: 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 The set of points form the

180-

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

Page 25: 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 The set of points form the

β

Instructions:

• Draw a cyclic quadrilateral

• Produce a side of the quadrilateral

•Measure angles and β

Page 26: 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 The set of points form the

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

the interior opposite angle

Page 27: 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 The set of points form the

Circle Circle PropertiesProperties

Part II tangent properties

Page 28: 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 The set of points form the

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

Page 29: 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 The set of points form the

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.

Page 30: 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 The set of points form the

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

are equalare equal

Page 31: 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 The set of points form the

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

Page 32: 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 The set of points form the

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

Page 33: 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 The set of points form the

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

Page 34: 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 The set of points form the

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

Page 35: 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 The set of points form the

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

Page 36: 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 The set of points form the

Circle Circle PropertiesProperties

Chord properties

Page 37: 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 The set of points form the

A

B

C

D

X

AX.XB=CX.XD

Triangle AXD is similar to triangle CXB hence

Page 38: 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 The set of points form the

A

B

C

D

X

AX.XB=CX.XD

Note: X is the crucial point in the formula

Page 39: 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 The set of points form the

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

Page 40: 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 The set of points form the

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

Page 41: 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 The set of points form the

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

Page 42: 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 The set of points form the

A

B

C

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

Page 43: 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 The set of points form the

A

B

C

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

Page 44: 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 The set of points form the

Quick Quick QuizQuiz

Page 45: 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 The set of points form the

a

40

a= 40

Page 46: 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 The set of points form the

b

40

b= 80C

Page 47: 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 The set of points form the

d

60 d= 120C

Page 48: 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 The set of points form the

f

55 f= 55C

Page 49: 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 The set of points form the

m=62

C62

m

Page 50: 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 The set of points form the

e

e= 90C

Page 51: 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 The set of points form the

x= 12

C102

102

12 cm

x cm

Page 52: 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 The set of points form the

k

70

k= 35C

Page 53: 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 The set of points form the

a

120

a= 50

10

Page 54: 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 The set of points form the

x100

x= 50C

Page 55: 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 The set of points form the

y y= 55C

35

Page 56: 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 The set of points form the
Page 57: 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 The set of points form the

Quick Quick QuizQuiz

Page 58: 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 The set of points form the

answer=A

10575

Which quadrilateral is concyclic?

A

B

C

100

110

20

140

Page 59: 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 The set of points form the

c

60 c =60C

Tangent

Page 60: 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 The set of points form the

g

g= 90C

Tangent

Page 61: 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 The set of points form the

h= 4C

4cm

h cm

Tangent

Tangent

Page 62: 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 The set of points form the

m

40

m =50C

Tangent

y =50

y

Page 63: 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 The set of points form the

a= 65C

50

Q

a

P

R

PQ, RQ are tangents

Page 64: 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 The set of points form the

n= 5

C

10

4

8

n

nx8=4x10

8n =40

n =5

Page 65: 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 The set of points form the

q= 25

C

10

4

q4q=10

2

4q=100

q=25

Page 66: 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 The set of points form the

x= 12

C

8

4

x

4(4+x)=82

4(4+x)=64

4+x=16

x=12

BA2=BC.BD

Page 67: 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 The set of points form the

k= 5

C

8m

k

3m

K2=32+42

K =5

Page 68: 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 The set of points form the