The Circle

Int 2

Monday 24 April 2023Monday 24 April 2023 11

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

The CircleThe Circle

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

Int 2


Aim of Today’s Lesson

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

Arc 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 arc


Int 2


Q. Find the circumference of the circle ?

SolutionSolution

C D3.14 20C

62.8C cm

10cm


Int 2


45 (3.14 12)360 o

oarc length

length 4.71arc cm


Arc lengthπD

Arc angle360o

=

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

6 cm45o

x

y

360o

connection

Int 2


60 (3.14 18)360 o

oarc length

length 9.42arc cm


Arc lengthπD

Arc angle360o

=

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


Arc lengthπD

Arc angle360o

=

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

10 m

100o

A

B

connection

260o

Int 2

24 Apr 202324 Apr 2023

Now try Ex 1

Ch6 MIA (page 60)


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

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.

Int 2


A

Bminor sector

major sector

Area of Sector in a Area of Sector in a circlecircle

Int 2


Q. Find the area of the circle ?

SolutionSolution2A r

23.14 10A 2314A cm

10cm


Int 2


245Area of Sector (3.14 6 )360 o

o

2Area Sector 14.14 cm

Area Sectorπr2

Sector angle360o

=

Find the area of the minor sector XY below ?

6 cm45o

x

y

360o

connection


Int 2


260Area Sector (3.14 9 )360 o

o

2 Sector 42.41Area cm

Area Sectorπr2

Sector angle360o

=

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


Sector Areaπr2

Sector angle360o

=

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

10 m

100o

A

B

connection

260o

Int 2

24 Apr 202324 Apr 2023

Now try Ex 2

Ch6 MIA (page 62)


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

Int 2


7.07 360(3.14 18)

o ox

45o ox


Arc lengthπD

Arc angle360o

=

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


Sector Areaπr2

Sector angle360o

=

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

10 mA

B

connection

xo

Int 2

24 Apr 202324 Apr 2023

Now try Ex 3

Ch6 MIA (page 64)

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

Int 2


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


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


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 180180 80 22 100

50

o o o

o o o

o o

o o

xx

xx


Int 2

24 Apr 202324 Apr 2023

Now try Ex 4

Ch6 MIA (page 66)


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.

1 2 102x y

Int 2


To understand some special properties

when a diameter bisects a chord.


Diameter symmetryDiameter symmetry

Int 2



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 have2 2 2

2 2 2

2 2 2

2

6 1010 6100 36 6464 8

a b caaa

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

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.


2 2 2

2 2 2

2

4 55 425 16 99 3

OM AM OAOMOMOMOM cm

AM is 8 ÷ 2 = 4Radius of the circle is 5cm.

CM = 3cm + radius

CM = 3 + 5 = 8cm

Int 2

24 Apr 202324 Apr 2023

Now try Ex 14.2

Ch14 (page 188)

Symmetry & Symmetry & Chords Chords

in Circlesin Circles

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 initially. How many are there after 9 hours.

5 1 0x y

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.


Int 2


Semi-circle angleSemi-circle angle

Tool-kit requiredTool-kit required

1.1. ProtractorProtractor

2.2. PencilPencil

3.3. RulerRuler

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.


Int 2


Mark three points Mark three points

1.1. Outside the circleOutside the circle

x xx

x xx x

x

x


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

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

Int 2


InsideCircumferenceOutside

x


xx

< 90o > 90o= 90= 90oo

Every angle in a semi-circle is a right angle

Online Demo

Int 2

24 Apr 202324 Apr 2023

Now try Ex 7

Ch6 MIA (page 71)

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

Int 2


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

radius at the point of contact.


Tangent lineTangent line

Int 2



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?

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.

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.


2 2 2

2 2 2

2

8 1010 8100 64 3636 6

a b caaaa

Int 2

24 Apr 202324 Apr 2023

Now try Ex 8

Ch6 MIA (page 73)

Tangent LinesTangent Lines

Int 2


Area is2A r

Summary of Circle Summary of Circle TopicTopic

Circumference isC D

Sector area2 angleSector = 360

o

ocentre r

Arc length is

length angleArc = 360

o

ocentre D

Diameter2D r

Radius12r 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 90o

Tangent touches circle at

one pointand make angle 90o with point of contact radius

Date post:	16-Mar-2016
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The Circle

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