S3 Friday, 17 April 2015Friday, 17 April 2015Friday, 17 April 2015Friday, 17 April 2015Created by Mr Lafferty1 Isosceles Triangles in Circles Right angle

S3

Tuesday, April 18, 2023Tuesday, April 18, 2023 Created by Mr Lafferty Created by Mr Lafferty 11

Isosceles Triangles in Circles Right angle in a Semi-Circle

The CircleThe Circlew

ww

.math

srevis

ion

.com

Tangent Line to a CircleDiameter Symmetry in a CircleCircumference of a CircleLength of an ARC of a CircleArea of a CircleArea of a SECTOR of a Circle

Summary of Circle Chapter

S3


When two radii are drawn to the ends of a chord, When two radii are drawn to the ends of a chord, An isosceles triangle is formed.An isosceles triangle is formed.

C

A B

ww

w.m

ath

srevis

ion

.co

mIsosceles triangles Isosceles triangles

in Circlesin Circles

S3


Special Properties of Isosceles TrianglesSpecial Properties of Isosceles Triangles

Two equal lengthsTwo equal lengths

Two equal anglesTwo equal angles

Angles in any triangle sum to 180Angles in any triangle sum to 180oo

ww

w.m

ath

srevis

ion

.co



S3


Q.Q.Find the angle xFind the angle xoo..

A

B

C

SolutionAngle at C is equal to:

Since the triangle is isosceleswe have

360 280 80o o o

xo

280o

2 80 180

180 80 2

2 100

50

o o o

o o o

o o

o o

x

x

x

x

ww

w.m

ath

srevis

ion

.co



S3


Tangent lineTangent line

A A tangent linetangent line is a line that is a line that touches a circle at touches a circle at only one point.only one point.

Which of theWhich of thelines are lines are tangent to tangent to the circle?the circle?

ww

w.m

ath

srevis

ion

.co

m

S3



The radius of the circle that touches the tangent The radius of the circle that touches the tangent line is called line is called the point of contact radius.the point of contact radius.

Special PropertySpecial Property

The point of contact radiusThe point of contact radiusis always perpendicular is always perpendicular

(right-angled)(right-angled)to the tangent line.to the tangent line.

ww

w.m

ath

srevis

ion

.co

m

S3



Q.Q.Find the length of the tangent line betweenFind the length of the tangent line between A and B.A and B.

A

B

8

10

C

SolutionRight-angled at A sinceAC is the radius at the pointof contact with the Tangent.

By Pythagoras Theorem we have2 2 2

2 2 2

2 2 2

2

8 10

10 8

100 64 36

36 6

a b c

a

a

a

a

ww

w.m

ath

srevis

ion

.co

m

S3


Diameter symmetryDiameter symmetry

A B

C

D

1. A line drawn through the centre of a circle through the midpoint a chord will ALWAYS cut

the chord at right-angles

2. A line drawn through the centre of a circle at right-angles to a chord willALWAYS bisect that chord.

3. A line bisecting a chord at right angles will ALWAYS pass through the centre of a circle.

O

S3


Q.Q.Find the length of the chord A and B.Find the length of the chord A and B.

A

B

6O

Solution

By Pythagoras Theorem we have

2 2 2

2 2 2

2 2 2

2

6 10

10 6

100 36 64

64 8

a b c

a

a

a

a

Diameter symmetryDiameter symmetry

4

Since yellow line bisect AB and passesthrough centre O, triangle is right-angle.

Radius of the circle is 4 + 6 = 10.

10

Since AB is bisected The length of AB is

2 8 16ABlength

ww

w.m

ath

srevis

ion

.co

m

S3


Main parts of the circle

radius

O

CircumferenceC D

Diameter2D r

ww

w.m

ath

srevis

ion

.co

mCircumference Circumference

of a circleof a circle

S3


Q. Find the circumference of the circle ?

SolutionSolution

C D 4C

12.56C cm

4cm

ww

w.m

ath

srevis

ion

.co



S3

Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1212

Q. The circumference of the circle is 60cm ? Find the length of the diameter and radius.

SolutionSolution

C D 60 D

60D cm

19 D cm

2D r

19 2r192

r

9.5r cm

ww

w.m

ath

srevis

ion

.co



S3


Q. What is an arc ?

A

B

AnswerAnswer

An arc is a fractionof the circumference.

minor arc

major arcww

w.m

ath

srevis

ion

.co

mlength of the length of the arc of a circlearc of a circle

S3


Q. Find the circumference of the circle ?

SolutionSolution

C D 20C

62.8C cm

10cm

ww

w.m

ath

srevis

ion

.co


S3


Q. Find the arc length for the semi-circle ?

Solution

C D length of circle circumf erencearc Fraction

31.4arc length cm

1

length ( )2

arc D180o

1

length ( 20)2

arc

ww

w.m

ath

srevis

ion

.co


S3


Q. Given part of the circle below. Find the arc length ?Solution

C D

1

( )4

arc length D

length 15.7arc cm

10cm length of circle circumf erencearc Fraction 90

o

1

( 20)4

arc length

ww

w.m

ath

srevis

ion

.co


S3


3 step process to calculating the length of an arc.

Step 1 Find the circumference of the circle.

Step 2 Find fraction of the whole circle.

Step 3 Find the arc length.

ww

w.m

ath

srevis

ion

.co


S3


Solution

C D

1

( 12)8

arc length

length 4.71arc cm

6cm

length of circle circumf erencearc Fraction

45o

Q. Part of a circle is given below. Find the arc length.

45 1f raction=

360 8

o

o

Fraction of

circle

ww

w.m

ath

srevis

ion

.co


S3


3 step process to calculating the length of an arc.

Step 1 Find the circumference of the circle.


Step 3 Find the arc length.

C D

at centre360o

angle

length of circle circumf erencearc Fraction

ww

w.m

ath

srevis

ion

.co


S3


Q. Find the length of the arc AB below ?

9 cm

60o

A

B

SolutionSolution

1 : C = D

= 18

= 56.52cm

Step

60 1 2 : Fraction=

360 6Step

1 3 : arc length= 56.52 9.42

6Step cm

ww

w.m

ath

srevis

ion

.co


S3


Q. The diameter of the circle is 60cm. Find area of the circle?

SolutionSolution

2A r60

302 2D

r cm

230A22827.43A cm

ww

w.m

ath

srevis

ion

.co

mThe Area of a The Area of a circlecircle

S3


Q. The area of a circle is 12.64 cm2.

Find its radius?SolutionSolution

2A r

2 12.64

4r cm

4 2r cm

212.64 r

ww

w.m

ath

srevis

ion

.co

mThe Area of a The Area of a circlecircle

S3


Q. What is an sector area ?

Sector area of a circleSector area of a circle

A

B

AnswerAnswerSector area is a fraction

of the area of the whole circle.

minor sector

major sectorww

w.m

ath

srevis

ion

.co

m

S3



Q. Find the area of the circle ?

SolutionSolution

2A r 210A

2314A cm

10cm

ww

w.m

ath

srevis

ion

.co

m

S3


Q. Find the sector area for the semi-circle ?

Solution

2A r of circle Area of circleSector Fraction

2157Sector cm


21( )

2Sector r180

o

21( 10 )

2Sector

10cm

ww

w.m

ath

srevis

ion

.co

m

S3


Q. Given part of the circle below. Find the sector area ?Solution

2A r

21( )

4Sector r

27.85Sector cm


c of circle Area of circleSe tor Fraction

21c ( 10 )

4Se tor

ww

w.m

ath

srevis

ion

.co

m

S3



3 step process to calculating sector area of a circle.

Step 1 Find the area of the circle.


Step 3 Find the sector area.

ww

w.m

ath

srevis

ion

.co

m

S3


Solution

2A r

1

( 36)8

Sector

2Sector 14.13cm

6cm


of circle area of circleSector Fraction

45o

Q. Part of a circle is given below. Find the sector area.

45 1f raction=

360 8

o

o

Fraction of

circle

ww

w.m

ath

srevis

ion

.co

m

S3



3 step process for calculating the sector area.

Step 1 Find the area of the circle.


Step 3 Find the sector area.

2A r

at centre360o

angle

ec of circle area of circleS tor Fraction

ww

w.m

ath

srevis

ion

.co

m

S3


Q. Find the area of the sector below ?


SolutionSolution

2

2

1 : A=

= 81

=254.34cm

Step r

60 1 2 : Fraction=

360 6Step

21 3 : arc length= 254.34 42.39

6Step cm

ww

w.m

ath

srevis

ion

.co

m

S3


Area is2A r

Summary of Circle Summary of Circle TopicTopic

Circumference is

C D

Sector area2 angle

Sector = 360

o

o

centrer

Arc length is

length

angleArc =

360

o

o

centreD

Diameter2D r

Radius12

r D

line that bisects a chord

1. Splits the chord into 2 equal halves.

2. Makes right-angle with the chord.

3. Passes through centre of the circle

Pythagoras TheoremSOHCAHTOA

Semi-circle angle is always 90

o

Tangent touches circle at

one pointand make angle 90

o with point of

contact radius

ww

w.m

ath

srevis

ion

.co

m

Documents

S3 Friday, 17 April 2015Friday, 17 April 2015Friday, 17 April 2015Friday, 17 April 2015Created by Mr Lafferty1 Isosceles Triangles in Circles Right angle