S3 Tuesday, 18 August 2015Tuesday, 18 August 2015Tuesday, 18 August 2015Tuesday, 18 August 2015Created by Mr Lafferty1 Isosceles Triangles in Circles Right

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Wednesday 19 April 2023Wednesday 19 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 Questionsw

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Q4. If a = 1 , b = 2 and c = 4

Find

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Q2. 30% of 200

( 25) ( 20)

5cm

2cm

3cm

4cm

2c - ab

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To identify isosceles triangleswithin 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

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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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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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Begin Maths in Action Book page 181

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1. (-5) - (-6) + 11

2. Calculate 15% of 60

3. What is the perimeter and area

of a square with length 4cm.

4. Find the value of x

3x - 9 = 27

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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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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


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ww

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Write down as many equations as you can

e.g. a + b = 11

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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 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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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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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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ww

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Q4. If x = -1 , y = 3 and z = 10

Find

Q3.

Q2. 15% of 280

( 22) ( 29)

10cm

3cm

5cm

7cm

2z - 2xy

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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

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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 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

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Diameter symmetryDiameter symmetryw

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Starter QuestionsStarter Questions

Q1.2 81Factorise x

Q2.12 4

Calculate 28 3

Q3. Calculate the area of the triangle

Q4.2Solve 2 3 0x x

12

8

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Circumference Circumference of a circleof a circle


To be able 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 4C

12.56C 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. Calculate

72Simplif y

360

Q2. Make D the subject of the equation.

Q4. Calculate

5( 2) 2 ( 3) x x a

C D

2 2117 36

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


To find and be able 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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Q. Find the circumference of the circle ?

SolutionSolution

C D 20C

62.8C cm

10cm

Maths in Action Book Exercise 1 Questions 1 & 2 ww

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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

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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

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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.

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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

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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

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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

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Starter Questions Starter Questions

Q1. Find all missing angles

Q2. List the prime numbers between 10 and 20.

Q3. Solve

Q4. Calculate the area of the shape.

52o

3( 2) 5( 6)t t

10cm

5cm

4cm

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The Area of a The Area of a circlecircle


To come up with and be able 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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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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Ex9.1 page 194 Q3-4

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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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Ex9.1 page 194 Q5 onwards

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Q3. Calculate

Q1. Factorise

2 51 25 8

Q2. Make h the subject of the equation.

Q4. The value of the standard deviation for petrolprices in Glasgow is 2.1 and 1.9 for Edinburgh.Comparing the two values write down a valid statement.

24 12 9 x x1 h

yt

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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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Q. What is an sector area ?


A

B

AnswerAnswerSector area is a fraction

of the area of the whole circle.

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Q. Find the area of the circle ?

SolutionSolution

2A r 210A

2314A cm

10cm

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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

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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

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3 step process to calculating sector area of a circle.

Step 1 Find the area of the circle.


Step 3 Find the sector area.

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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

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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

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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

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Q1.Q1. Find all the missing anglesFind all the missing angles

Q2.Q2. CalculateCalculate

2 2 where a = 4 and b = (-5)a b

Q3. Q3. The cost of a new computer is £1000 +vat. If the vat is The cost of a new computer is £1000 +vat. If the vat is charged at 12% what is the total cost.charged at 12% what is the total cost.

Q4. Q4. The cost of a bag of sugar is £1.12. How much 50 bags The cost of a bag of sugar is £1.12. How much 50 bags cost.cost.

115o

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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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Summary of Circle TopicSummary of Circle Topic

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

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Documents

S3 Tuesday, 18 August 2015Tuesday, 18 August 2015Tuesday, 18 August 2015Tuesday, 18 August 2015Created by Mr Lafferty1 Isosceles Triangles in Circles Right