1178858588 2006 Mathematics Extension 1 Notes

8/3/2019 1178858588 2006 Mathematics Extension 1 Notes

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Circle Properties

Part I


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

a plane

The set of points form the .Circumference


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

the.Radius


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A is a straight line segment joiningtwo points on the circle

chord


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A chord that passes through the centre is a

.diameter


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A is a straight line thatcuts the circle in two points

secant


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An arc is part of the circumference of a circle

Major arc

Minor arc


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A is part of the circlebounded by two radii and an arc

sector

Minorsector

majorsector


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A is part of the circlebounded by a chord and an arc

segment

Minor

segment

majorsegment


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

AB

O

Subtendsmeans to extend under or to beopposite to


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Instructions:

Draw a circle

Draw two chords of equal length

Measure angles AOB and DOC

A

B

C

D

O

What do you notice?


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Equal chords subtend equal angles at the centre


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Conversely

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


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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?


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Instructions:

select an arc AB

subtend the arc AB to the centre O


Measure angles AOB and ACB

B

O

A

C

What do you notice?


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2

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

at the circumference


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Instructions:

select an arc AB

select two points C, D on the circumference


subtend the arc AB to a point D on the circumference

Measure angles ACB and ADB

B

O

A

C

D


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Instructions:

select an arc AB

select two points C, D 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?


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Angles subtended at the circumference by thesame arc are equal


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Instructions:

Draw a circle and its diameter

subtend the diameter to a point on the circumferenceMeasure ACB

C

B

What do you notice?

A


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An angle in a semicircle isa right angle


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Instructions:

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

Measure all four angles

What do you notice?


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180-

The opposite angles of a cyclic quadrilateralare supplementary

180-


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180-

If the opposite angles of a quadrilateral aresupplementary the quadrilateral is cyclic


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Instructions:

Draw a cyclic quadrilateral

Produce a side of the quadrilateral

Measure angles and


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

opposite angle


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Circle Properties

Part II tangent properties


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


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Tangent to a circle is perpendicular to the

radius drawn from the point of contact.


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Tangents to a circle from an exterior point are equal


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When two circles touch, the line through their centres

passes through their point of contact

Point of contact

External Contact


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When two circles touch, the line through their centres

passes through their point of contact

Point of contact

Internal Contact


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The angle between a tangent

and a chord through the point of contact

is equal to the angle in the alternate segment


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The square of the length of the tangent

from an external point is equal to

the product of the intercepts of the secant

passing through this point

A

B

BA2

=BC.BD

C

D

B=external point


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The square of the length of the tangent

from an external point is equal to

the product of the intercepts of the secant

passing through this point

A

B

BA2

=BC.BD

C

D

Note: B is the crucial point in the formula


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Circle Properties

Chord properties


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A

B

C

D

X

AX.XB=CX.XD

Triangle AXD is similar to triangle CXBhence


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A

B

C

D

X

AX.XB=CX.XD

Note: X is the crucial point in the formula


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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


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A

B

C

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


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A

B

C

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


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A

B

C

Equal chords are equidistant from the centre of the circle


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A

B

C

Conversley: Chords that are equidistant from the centreare equal


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Quick

Quiz


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a

40a= 40


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b

40

b= 80

C


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d

60 d= 120

C


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f

55 f= 55

C


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m=62

C

62m


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e

e= 90

C


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x= 12

C

102102

12 cm

x cm


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k

70

k= 35

C


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a

120a= 50

10


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x100

x= 50

C


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y y= 55

C

35


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QuickQuiz

Which quadrilateral is concyclic?


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answer= A

10575

Which quadrilateral is concyclic?A

B

C

100

110

20

140


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c

60 c= 60

C


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g

g= 90

C

4cm


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h= 4

C

4cm

h cm

Tan

gent


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m

40

m=50

C

y=50

y

Q


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a= 65

C

50 Q

a

P

R

PQ, RQ are tangents


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n= 5

C

10

4

8

n

nx8=4x10

8n =40

n =5


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q= 25

C

10

4

q 4q=102

4q=100

q=25

BA2

BC BD


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x= 12

C

8

4

x

4(4+x)=82

4(4+x)=64

4+x=16

x=12

BA2=BC.BD


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k= 5

C

k

K2=32+42

K =5


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1178858588 2006 Mathematics Extension 1 Notes

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