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National 5

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

Isosceles Triangles in Circles Right angle in a Semi-Circle

The CircleThe Circlew

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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

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Starter QuestionsStarter Questions

Q3.

Q1. True or false

2Does 8 12 f actorise to (x - 6)(x - 2)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.

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

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We are learning to identify isosceles triangles

within a circle.

Aim of Today’s Lesson

Isosceles Triangles Isosceles Triangles in Circlesin Circles

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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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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

Solution

Angle 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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Maths in Action Ex 2.1 page 181

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Q3.

Q1. Explain how we 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. Explain why the originally price was £200.

5( 2) 20x

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Semi-circle angleSemi-circle angle

We are learning to find the angle in a semi-circle made

by a triangle with hypotenuse equal to the diameter and the two smaller

lengths meeting at the circumference.


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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

x


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

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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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InsideCircumferenceOutside

x


x

x

< 90o

> 90o

= 90= 90oo

Begin Maths in Action Book page 182

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DEMO

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Starter QuestionsStarter Questionsw

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Write down as many equations as you can

e.g. a + b = 11

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We are learning to understand what a tangent line is and 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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DEMODEMO

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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 have

2 2 2a b c

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2 2 28 10a

2 2 210 8a 2 100 64 36a

36 6a

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Q2. 2Factorise 4 9x

Q1. Using FOIL multiply out (x2 + 4x - 3)(x + 1)

Q3. I want to make 15% profit on a computer I bought for £980. How much must I sell it for.w

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Using a multiplication table expand out (x2 + 4x - 3)(x + 1)

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We are learning to understand some special properties

when a diameter bisects a chord.


Diameter symmetryDiameter symmetryw

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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

DEMO

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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

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

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2 2 26 10a

2 2 210 6a

2 100 36 64a

64 8a

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Maths in Action

Ex 5.1 & Ex 5.2 page 187

Diameter symmetryDiameter symmetryw

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Q1.2 81Factorise x

Q2.12 4 4

True or f alse 28 3 7

Q3. Explain why the area of the triangle is 48m2

Q4.2Solve 2 3x x

12m

8m

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Circumference Circumference of a circleof a circle


We are learning to use the formula for calculating

the circumference of a circle

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Main parts of the circle

radius

O

CircumferenceC D

Diameter2D r

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of a circleof a circle

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Q. Find the circumference of the circle ?

SolutionSolution

C D 8C

25.12C cm

4cm

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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

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Now it’s your turn !


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Q3.

Q1. True or false

72Simplif y

360

Q2. Using the balancing method rearrange into D =

Q4. Calculate

5(2 1) 2(5 1) x x

C D

2 2117 36

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


We are learning to use the formula for calculating the length of an arc.

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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

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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

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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

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

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Q3.

Q1. True or false

2Does 4 4 f actorise (x - 2)(x - 2)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.

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

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The Area of a The Area of a circlecircle


We are learning to use the formula for calculating

the area of a circle

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23

4

12

3

456

7

8

1 56

78

If we break the circleinto equal sectors

And lay them out side by side We get very close

to a rectangle.

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mThe Area of a The Area of a circlecircle

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12 3

45

67

8

thinner and thinnersectors

If we cut the sectors Thinner and thinner then we get closer and closer

to a rectangle. Hence we can represent the area of a circle

by a rectangle.

r

r

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r

r

2Area of a circle r

Area of a rectangle l b 2Area of a rectangle r r r

But the area inside this rectangle is also the area of the circle

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Q. Find the area of the circle ?

SolutionSolution

2A r 24A

250.26A cm

4cm

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The Area The Area of a circleof a circle


Begin Maths in Action Book

Ex9.1 page 194 Q1-2

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

SolutionSolution

2A r60

302 2D

r cm

230A22827.43A cm

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Maths in Action

Ex9.1 page 194

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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

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Begin Maths in Action Book

Ex9.1 page 194 Q5 onwards

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Q3. Calculate

Q1. Find the missing numbers

2 51 25 8

Q2. Using the balancing method rearrange into x =

24 12 9 (2 ?)(? ?) x x x x

1 y x

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Sector area of a circleSector area of a circle


We are learning to use the formula for calculating the sector of an circle.

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A

B

major sector

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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

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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

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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

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Sector area of a circleSector area of a circle



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Q3.

Q1. Using the balancing method rearrange into x =

2Factorise 3 2 x x

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

12 10

2x y

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Area is2A r

Summary of Circle Summary of Circle TopicTopic

Circumference is

C D

Sector area

sector2

angleArea =

360

o

o

centrer

Arc length is

lengthArc angle

= D 360

o

o

centre

Diameter

2D rRadiu

s12

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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Summary of Circle TopicSummary of Circle Topic

Maths in Action Book Page 199Maths in Action Book Page 199

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