Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty1 Length of Arc in a Circles Area of

Int 2

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

Length of Arc in a Circles Area of a Sector in a Circle

The CircleThe Circlew

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Symmetry in a Circle

Diameter & Right Angle in a CircleSemi-Circle & Right Angle in a CircleTangent & Right Angle in a Circle

Summary of Circle Chapter

Mix Problems including Angles

Int 2


Starter QuestionsStarter Questions

Q3.

Q1. Simplify

2Factorise 8 12 x x

Q2. How many degrees in one eighth of a circle.

Q4. After a discount of 20% an iPod is £160. How much was it originally.

5( 2) 2 ( 3) x x a

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Int 2


Aim of Today’s Lesson

To find and be able to use the formula for calculating the length of an arc.

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mArc length of a Arc length of a

circlecircle

Int 2


Q. What is an arc ?

A

B

AnswerAnswer

An arc is a fractionof the circumference.

minor arc

major arcww

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circlecircle

Int 2


Q. Find the circumference of the circle ?

SolutionSolution

C D3.14 20C

62.8C cm

10cm

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circlecircle

Int 2


45 (3.14 12)

360

o

oarc length

length 4.71arc cm

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circlecircle

Arc lengthπD

Arc angle

360o=

Q. Find the length of the minor arc XY below ?

6 cm45

o

x

y

360o

connection

Int 2


60 (3.14 18)

360

o

oarc length

length 9.42arc cm

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circlecircle

Arc lengthπD

Arc angle

360o=

Q. Find the length of the minor arc AB below ?

9 cm

60o

A

B

connection

Int 2


260 (3.14 20)

360

o

oarc length

length 45.38arc cm

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circlecircle

Arc lengthπD

Arc angle

360o=

Q. Find the length of the major arc PQ below ?

10 m

100o

A

B

connection

260o

Int 2

11 Apr 202311 Apr 2023 Created by Mr. Created by Mr. [email protected]@mathsrevision.com

Now try Ex 1

Ch6 MIA (page 60)

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Arc length of a Arc length of a circlecircle

Int 2



Q3.

Q1. Solve

2Factorise 13 42 x x

Q2. How many degrees in one tenth of a circle.

Q4. After a discount of 40% a Digital Radio is £120. How much was it originally.

5( 2) 20x

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Int 2


Sector area of a circleSector area of a circle


To find and be able to use the formula for calculating the sector of an circle.

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Int 2


A

B

minor sector

major sector

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circlecircle

Int 2


Q. Find the area of the circle ?

SolutionSolution

2A r23.14 10A

2314A cm

10cm

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circlecircle

Int 2


245Area of Sector (3.14 6 )

360

o

o

2Area Sector 14.14 cm

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Area Sectorπr2

Sector angle

360o=

Find the area of the minor sector XY below ?

6 cm45

o

x

y

360o

connection

Area of Sector in a Area of Sector in a circlecircle

Int 2


260Area Sector (3.14 9 )

360

o

o

2 Sector 42.41Area cm

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Area Sectorπr2

Sector angle

360o=

Q. Find the area of the minor sector AB below ?

9 cm

60o

A

B

connection


Int 2


2260Sector Area (3.14 10 )

360

o

o

2Area Sector 226.89 cm

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circlecircle

Sector Areaπr2

Sector angle360o

=

Q. Find the area of the major sector PQ below ?

10 m

100o

A

B

connection

260o

Int 2


Now try Ex 2

Ch6 MIA (page 62)

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Int 2



Q3.

Q1. Find the gradient and y - intercept

2Factorise 4 9x

Q2. Expand out (x2 + 4x - 3)(x + 1)

Q4. I want to make 15% profit on a computer I bought for £980. How much must I sell it for.

2 4 16y x

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Int 2


7.07360

(3.14 18)

o ox

45o ox

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circlecircle

Arc lengthπD

Arc angle

360o=

Q. The arc length is 7.07cm. Find the angle xo

9 cm

xo

A

B

connection

Int 2


o2

235.62x 360

(3.14 10 )

o

ox 270 o

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circlecircle

Sector Areaπr2

Sector angle360o

=

Q. Find the angle given area of sector AB is 235.62cm2 ?

10 m

A

B

connection

xo

Int 2


Now try Ex 3

Ch6 MIA (page 64)

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Mixed ProblemsMixed Problems

Int 2



Q3.


2Factorise 4 4 x x

Q2. Expand out (x + 3)(x2 + 40 – 9)

Q4. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for.

2 2 20x y

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Int 2


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To identify isosceles triangleswithin a circle.


Isosceles triangles Isosceles triangles in Circlesin Circles

Int 2


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

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mIsosceles triangles Isosceles triangles

in Circlesin Circles

Online Demoxo xo

Int 2


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

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Int 2


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

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Int 2


Now try Ex 4

Ch6 MIA (page 66)

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Isosceles triangles Isosceles triangles in Circlesin Circles

Int 2



Q3.


2Factorise 3 2 x x

Q2. Expand out 2(x - 3) + 3x

Q4. Car depreciates by 20% each year. How much is it worth after 3 years.

12 10

2x y

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Int 2


To understand some special properties

when a diameter bisects a chord.


Diameter symmetryDiameter symmetryw

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Int 2


Diameter symmetryDiameter symmetry

A B

C

D

1. A line drawn through the centre of a circle and 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 will

ALWAYS bisect (cut equally in 2) that chord.

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

o

Int 2


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


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

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Int 2

C



A B

D

o

M

Find the length of OM and CM. Radius of circle is 5cm,

AB is 8cm and M is midpoint of AB.

By Pythagoras Theorem we have2 2 2

2 2 2

2 2 2

2

4 5

5 4

25 16 9

9 3

OM AM OA

OM

OM

OM

OM cm

AM is 8 ÷ 2 = 4

Radius of the circle is 5cm.

CM = 3cm + radius

CM = 3 + 5 = 8cm

Int 2


Now try Ex 5

Ch6 MIA (page 68)

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Symmetry & Symmetry & Chords Chords


Int 2



Q3.


2Factorise 2d g dg

Q2. Expand out (x + 3)(x + 4)

Q4. Bacteria increase at a rate of 10% per hour. If there were 100 bacteria initial. How many are there after 9 hours.

5 1 0x y

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Int 2


Semi-Circle AngleSemi-Circle Angle

To find the angle in a semi-circle made by a triangle with hypotenuse

equal to the diameter and the two smallerlengths meeting at the circumference.


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Int 2


Semi-circle angleSemi-circle angle

Tool-kit requiredTool-kit required

1.1. ProtractorProtractor

2.2. PencilPencil

3.3. RulerRuler

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Int 2


1.1. Using your pencil trace roundUsing your pencil trace roundthe protractor so that you havethe protractor so that you have semi-circle.semi-circle.

2.2. Mark the Mark the centre of centre of the the semi-circle.semi-circle.

You should have You should have something like this.something like this.

Semi-circle angleSemi-circle anglew

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Int 2


Mark three points Mark three points

1.1. Outside the circleOutside the circle

x xx

x x

x x

x

x


2. On the 2. On the circumferencecircumference3. Inside the circle3. Inside the circle

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Int 2


For each of the points For each of the points

Form a triangle by drawing aForm a triangle by drawing aline from each end of the line from each end of the diameter to the point.diameter to the point.Measure the angle at the Measure the angle at the various points.various points.

x

x

x


Log your results in a table. Log your results in a table. InsideCircumferenc

eOutside

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Int 2


InsideCircumferenceOutside

x


x

x

< 90o

> 90o

= 90= 90oo

Begin Maths in Action Book page 182

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Online Demo

Int 2


Now try Ex 7

Ch6 MIA (page 71)

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Angles in a Semi-Angles in a Semi-CircleCircle

Int 2



Q3.


2Factorise 36x

Q2. Expand out (a - 3)(a2 + 3a – 7)

Q4. I want to make 60% profit on a TV I bought for £240. How much must I sell it for.

8 2 16x y

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Int 2


To understand what a tangent line isand its special property with the

radius at the point of contact.


Tangent lineTangent linew

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Int 2


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?

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Int 2



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.

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Online Demo

Int 2



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

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Int 2


Now try Ex 8

Ch6 MIA (page 73)

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Tangent LinesTangent Lines

Int 2


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

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Documents

Int 2 Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Sunday, 24 August 2014Created by Mr Lafferty1 Length of Arc in a Circles Area of