The Circle

Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 11

The CircleThe Circlew

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Finding an ARC length

Finding the area of a SECTOR

Finding the centre angle given ARC length

Finding the centre angle given SECTOR area

Mixed questions

Exam questions

Isosceles Triangles in Circles

Right angle in a Semi-Circle

Tangent Line to a Circle

The Tangent Kite

Level 4


Starter QuestionsStarter Questions

Q1. True or false

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

Q3. After a discount of 20% an iPod is £160. How much was it originally.

25( 2) 2 ( 3) 2 11 10 x x x x x

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Level 4


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To identify isosceles triangleswithin a circle.

Aim of Today’s Lesson

Isosceles triangles Isosceles triangles in Circlesin Circles

Level 4


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

xo xoDemo

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Special Properties of Isosceles Triangles

Two equal lengths

Two equal angles

Angles in any triangle sum to 180o

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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

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2 80 180o o ox

2 100o ox

50o ox

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Special Properties of Isosceles Triangles

Two equal lengths

Two equal angles

Angles in any triangle sum to 180o

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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 2a b c


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

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2 2 26 10a

2 2 210 6a

2 100 36 64a

64 8a

Level 4

20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.

Now try N4+ TJEx18.1

Ch18 (page 139)

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Q1. Explain how we solve

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

Q3. After a discount of 40% a Digital Radio is £120. Explain why the originally price was £200.

5( 2) 20x

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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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National 5


Semi-circle angleSemi-circle angle

Tool-kit requiredTool-kit required

1.1. ProtractorProtractor

2.2. PencilPencil

3.3. RulerRuler

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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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Mark three points Mark three points

1.1. Outside the circleOutside the circle

x xx

x x

x x

x

x2. On the 2. On the

circumferencecircumference3. Inside the circle3. Inside the circle

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mSemi-Circle AngleSemi-Circle Angle

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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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Demo

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InsideCircumferenceOutside

xx

x

< 90o

> 90o

= 90= 90oo

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Demo

Circle Circumference Angle - Investigation Worksheet

Outer Circumference

Inner

Circle 1

Circle 2

Circle 3

Level 4



Ch18 (page 141)

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Semi-Circle AngleSemi-Circle Angle

Level 4


Starter QuestionsStarter Questionsw

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Write down as many equations as you can

e.g. a + b = 11

Level 4


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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Tangent lineTangent line

A A tangent line tangent 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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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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Demo

Level 4

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



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

Solution

By Pythagoras Theorem we have 2 2 2a b c

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2 2 28 10a

2 2 210 8a 2 100 64 36a 36 6a

Level 4



Ch18.3 (page 143)

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Tangent LineTangent Line

National 5



Q1. True or false

Q2. Expand out (x + 3)(x – 2)

26 8 2 (4 4) x x x x

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National 5

Tangent KiteTangent Kite

20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww

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1.1. Know the properties of Know the properties of tangent kites.tangent kites.

We are learning properties of a tangent kite.

2.2. Be able to calculate Be able to calculate lengths and angles related lengths and angles related to a tangent kite.to a tangent kite.

20 Apr 2023 Compiled by Mr. Lafferty Maths Dept.

Tangent Kite

A tangent kite has two right-angles

A tangent kite is made up of two congruent triangles


Tangent Kite

Find all angles in the tangent kite.

42o

90o

90o

138o

A

B

C


Tangent KiteFind the area of the circle.

40cm50

cm

Using Pythagoras Theorem

r

r2 = 502 - 402

r2 = 2500 - 1600

r2 = 900

r = 30cm√

A = πr2

A = π(30)2

A = 2827cm2

Level 4



Ch18.3 (page 145)

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Tangent LineTangent Line

National 5



Q1. True or false

Q2. Expand out (x + 3)(x2 + 40 – 9)

Q3. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for.

24 2 2 (2 1) x x x x

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Arc length of a circleArc length of a circle


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1.1. Know the arc length Know the arc length formula.formula.

We are learning to use the arc length formula.

2.2. Be able to calculate arc Be able to calculate arc length showing length showing appropriate work.appropriate work.

National 5


Q. What is an arc ?

A

B

AnswerAnswer

An arc is a fractionof the circumference.

minor arc

major arcww

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circlecircle

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Q. Find the circumference of the circle ?

SolutionSolution

C D 20C

62.8C cm

10cm

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circlecircle

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45

( 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

DEMO

National 5


60

( 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

National 5


260

( 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

P

Q

connection

260o


Now try N5 TJEx13.1

Ch13 (page 126)

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Arc length of a Arc length of a circlecircle

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Q1. True or false

Q2. Expand out (x + 3)(x2 + 40 – 9)

Q3. I want to make 30% profit on a DVD player I bought for £80. How much must I sell it for.

24 9 (2 3)(2 3) x x x

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Sector area of a circleSector area of a circlew

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1.1. Know the sector formula.Know the sector formula.We are learning how to find the area of a sector.

2.2. Be able to calculate sector Be able to calculate sector area showing appropriate area showing appropriate work.work.

Nat 5

National 5


A

B

major sector

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mArea of Sector in a Area of Sector in a

circlecircle

minor sector

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Q. Find the area of the circle ?

SolutionSolution

2A r 210A

2314A cm

10cm

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circlecircle

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245Area of Sector ( 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

DEMO

National 5


260Area Sector ( 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

A

B

connection

Area of Sector in a Area of Sector in a circlecircle

60o

National 5


2260Sector Area ( 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

P

Q

connection

260o


Now try N5 TJEx13.2

Ch13 (page 127)

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Q3.

Q1. Using the balancing method rearrange into x =

2Factorise 3 9x x

Q2. True or false 2(x - 3) + 3x = 5x - 6

12 10

2x y

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Mixed QuestionMixed Questionw

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1.1. Identify appropriate Identify appropriate formula to use.formula to use.

We are doing mixed questions on the circle.

2.2. Be able to calculate sector Be able to calculate sector / arc showing appropriate / arc showing appropriate work.work.

Nat 5


Now try N5 TJEx13.3

Ch13 (page 128)

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Rearrange the formula below into xo =

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o

o

Al xD 360

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1.1. Know formula.Know formula.We are learning how to find the centre angle given the arc length.

2.2. Be able to calculate centre Be able to calculate centre angle showing appropriate angle showing appropriate work.work.

Nat 5

Centre Angle (Arc Centre Angle (Arc length)length)

National 5


o o4.71x 360

12

o ox 45

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Arc lengthπD

Arc angle

360o=

Find the centre angle xo given the arc length is 4.71cm2.

6 cm45

o

x

yconnection



Now try N5 TJEx13.4

Ch13 (page 129)

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Rearrange the formula below into xo =

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o o

2 o

A S x

r 360

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Centre Angle (ACentre Angle (AooS)S)w

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1.1. Know formula.Know formula.We are learning how to find the centre angle given the sector area.

2.2. Be able to calculate centre Be able to calculate centre angle showing appropriate angle showing appropriate work.work.

Nat 5

National 5


o o2

42.41x 360

9

o ox 60

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Area Sectorπr2

Sector angle

360o=

Find the angle xo given that the sector area is 42.41cm2.

9 cm

A

B

connection

Centre Angle (ACentre Angle (AooS)S)

xo

National 5


o o2

51.3x 360

7

o ox 120

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Area Sectorπr2

Sector angle

360o=

Find the angle xo given that the sector area is 51.3mm2.

7 mm

A

B

connection


xo


Now try N5 TJEx13.5

Ch13 (page 130)

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Nat 5


3 marks

The Circle

Documents

Transcript of The Circle