Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty1 Circles Revision of Angle Properties Angles in

Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 11

CirclesCircles

Revision of Angle Properties

Angles in a semi-circle

ww

w.m

ath

srevis

ion

.com

Pythagoras Theorem

(SOH)(CAH)(TOA)

Tangent line on a circle

National 4

Isosceles Triangles in Circles


Starter QuestionsStarter Questionsw

ww

.math

srevis

ion

.com

o o o

1. Find the length the hypotenuse.

2. Find sin(c ), cos(c ) and tan(c ) f or triangle in Q1

3. What is the name of the shape with this net.

4. Calculate the value of the missing angle. 34o

A

B

C

6

8co

National 4

18 Apr 202318 Apr 2023 Created by Mr. LaffertyCreated by Mr. Lafferty

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. To know the basic To know the basic properties for angles.properties for angles.

1. To review angle properties.

2.2. Solve problems using Solve problems using properties.properties.

ww

w.m

ath

srevis

ion

.com

RevisionRevisionAngle PropertiesAngle Properties

National 4


120o 95

o

ww

w.m

ath

srevis

ion

.com

Angles round a pointAdd up to 360

o

115o

Two angles making astraight line add to 180

o

angles opposite each otherat a cross are equal.

34o 34

o

3 angles in a triangle ALWAYSadd up to 180

o.

50o

40o

65o

90o146o

146o

145o


National 4

18 Apr 202318 Apr 2023 Created by Mr. LaffertyCreated by Mr. Laffertyww

w.m

ath

srevis

ion

.com

Two angles in a isoscelesAre equal

ALL angles in an equilateral triangle are 60

o

d = 115oa

o

co

bo

go

fo

ho

eo

is corresponding to d and must be 115

o

is opposite to d and must be 115o

is must be 65o (straight line)

is alternate to c and must also be 65o

hbce


National 4


Now try Ex 1Ch12 (page 135)Copy out shapes

ww

w.m

ath

srevis

ion

.com


National 4



ww

.math

srevis

ion

.com 1. Find the length the hypotenuse.

2. Find 2x - 4 = 56


4. Calculate the value of the missing angle. 136o

A

B

C

3

4

National 4



1.1. To identify isosceles To identify isosceles triangles with circles.triangles with circles.

1. To investigate isosceles triangles with circles.

2.2. Solve problems using Solve problems using angle properties.angle properties.

ww

w.m

ath

srevis

ion

.com


National 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

ww

w.m

ath

srevis

ion

.co

mIsosceles triangles Isosceles triangles

in Circlesin Circles

xo xoDEMO


Special Properties of Isosceles Triangles

Two equal lengths

Two equal angles

Angles in any triangle sum to 180o

ww

w.m

ath

srevis

ion

.co




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

ww

w.m

ath

srevis

ion

.co



2 80 180o o ox

2 100o ox

50o ox


Now try Ex 1Ch12 (page 137)Copy out shapes

ww

w.m

ath

srevis

ion

.com


National 4



ww

.math

srevis

ion

.com

o o o

1. Find the length the hypotenuse.

2. Find sin(b ), cos(b ) and tan(b ) f or triangle in Q1


4. Calculate the value of the missing angle.

A

B

C

5

12

bo

27o

National 4



1.1. To know the special To know the special angle in a semi-circle.angle in a semi-circle.

1. To investigate the special angle in a semi-circle.

2.2. Solve problems using Solve problems using the special property.the special property.

ww

w.m

ath

srevis

ion

.com

Angles in a Semi CircleAngles in a Semi CircleNational 4

National 4


Tool-kit required

1. Protractor

2. Pencil

3. Ruler

ww

w.m

ath

srevis

ion

.co

mAngles in a Semi CircleAngles in a Semi Circle

National 4


1. Using your pencil trace roundthe protractor so that you have semi-circle.

2. Mark the centre of the semi-circle.

You should have something like this.

ww

w.m

ath

srevis

ion

.co


National 4


Mark three points

1. Outside the circle

x xx

x x

x x

x

x

2. On the circumference

3. Inside the circle

ww

w.m

ath

srevis

ion

.co


National 4


For each of the points

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

x

x

x

Log your results in a table. InsideCircumferenc

eOutside

ww

w.m

ath

srevis

ion

.co


National 4


InsideCircumferenceOutside

xx

x

< 90o

> 90o

= 90= 90oo

ww

w.m

ath

srevis

ion

.co

m

DEMO

Angles in a Semi CircleAngles in a Semi Circle


KeyPoint for Angles in a Semi-circle

Angles in a Semi-CircleAngles in a Semi-Circle

A triangle APB inscribed within a semicircle with hypotenuse equal to the diameterwill ALWAYS be right angled at P on the

circumference.

Remember - Angles in any triangle sum to 180o

ww

w.m

ath

srevis

ion

.co

m

P

A B

National 4


Hints

ww

w.m

ath

srevis

ion

.co

m

Example 1 : Sketch diagram and find all the missing angles.

70o

Look for right angle triangles43o

20o

47o

Remember ! Angles in any triangle

sum to 180o

Angles in a Semi-CircleAngles in a Semi-CircleNational 4


ww

w.m

ath

srevis

ion

.co

m

Example 2 : Sketch the diagram.


C

A B

D

25o

60o

(a) Right down two right angle triangles

(a) Calculate all missing angles.

E

National 4


Now try Ex 2Ch12 (page 138)Sketch shapes

ww

w.m

ath

srevis

ion

.com

Angles in a Semi-CircleAngles in a Semi-CircleNational 4



ww

.math

srevis

ion

.com

2 21. Calculate 8 .

2. An insurance company sells contents insurance

at a rate of 3.50 per £ 500 of value.

I want to insure a DVD player £ 350, TV £ 1850 and

games console £ 300. How

+ 10

much is the premium.

National 4



1.1. To know when to use To know when to use Pythagoras Theorem in Pythagoras Theorem in a circle.a circle.

1. To explain how we can use Pythagoras Theorem to calculate length within a circle.

2.2. Solve problems using Solve problems using Pythagoras Theorem.Pythagoras Theorem.

ww

w.m

ath

srevis

ion

.com

Angles in a Semi CircleAngles in a Semi CirclePythagoras TheoremNational 4


ww

w.m

ath

srevis

ion

.co

mAngles in a Semi-CircleAngles in a Semi-Circle

Pythagoras Theorem

We have been interested in right angled triangles withina semi-circle. Since they are right angled we can usePythagoras Theorem to calculate lengths.

P

A B

4cm3cm

d cm

Example 1 : Calculate the value of d

2 2 2d AP BP ( ) ( )2 2 24 3d 2 25d 25 5d cm

5cm

National 4


ww

w.m

ath

srevis

ion

.co


Pythagoras Theorem

Y

X Zcm

8cm

10 cm

Example 2 : Calculate the length of XY2 2 2XZ XY YZ ( ) ( ) ( )2 2 210 8XY ( ) ( ) ( )2 2 210 8XY ( )

36 6XY cm

2 100 64 36XY ( )

6cm

National 4


Now try Ex 3Ch12 (page 140)

Q1 to Q9Sketch shapes

ww

w.m

ath

srevis

ion

.com


Pythagoras TheoremNational 4



ww

.math

srevis

ion

.com 1. Solve 2y + 9 = 26

2. An insurance company sells contents insurance

at a rate of £ 3 per £ 200 of value.

I want to insure a DVD player £ 700, TV £ 1 200 and

games console £ 300. How much is the premium

if you get a 10% discount f or content value over

£ 2 000.

SOHCAHTOANational 4



1.1. To know when to use To know when to use Trigonometry (STrigonometry (SOOHCHCAAHTHTOOA) A) to calculate length and to calculate length and

angles angles within a circlewithin a circle..

1. To explain how we can use Trigonometry (SOHCAHTOA) to calculate length and angles within a circle.

2.2. Solve problems using Solve problems using Trigonometry (STrigonometry (SOOHCHCAAHTHTOOA) A) ..

ww

w.m

ath

srevis

ion

.com

Angles in a Semi CircleAngles in a Semi CircleSOHCAHTOANational 4


ww

w.m

ath

srevis

ion

.co

m

National 4


We have been interested in right angled triangles withina semi-circle. Since they are right angled we can useSOHCAHTOA to calculate lengths and angles.

Example 1 : Calculate the value of angle xo

o O 3tan = =

A 4x

1 3

4tanox

36 9ox .

P

A B

4cm3cm

d cm

xo=36.9o

SOHCAHTOA


ww

w.m

ath

srevis

ion

.co


B

A C67.4o

13 cm

Example 2 : Calculate the length of AB

13cos o A AB

xH

13 67 4cos . oAB

5AB cm

5cm

SOHCAHTOANational 4

National 4


Now try Ex 3Ch12 (page 140)

Q10 onwardsSketch shapes

ww

w.m

ath

srevis

ion

.com


SOHCAHTOA

National 4

If a = 7 b = 4 and c = 10

Write down as many equations as you can

Starter QuestionsStarter Questions

Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3434ww

w.m

ath

srevis

ion

.com

e.g. a + b + c = 21

National 4



1.1. To understand what a To understand what a tangent line is.tangent line is.

1. To explain how what a tangent

line is and its special property with the radius at the point of contact.

2.2. Solve problems using the Solve problems using the tangent property.tangent property.

ww

w.m

ath

srevis

ion

.com

Angles in a Semi CircleAngles in a Semi CircleTangent Line

National 4


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?

ww

w.m

ath

srevis

ion

.co


Tangent Line

National 4


The radius of the circle that touches the tangent line is called the point of contact radius.

Special Property

The point of contact radiusis always perpendicular

(right-angled)to the tangent line.

ww

w.m

ath

srevis

ion

.co


Tangent Line

DEMO

National 4


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

ww

w.m

ath

srevis

ion

.co


Tangent Line

National 4


Now try Work Sheet

Sketch shapes

ww

w.m

ath

srevis

ion

.com


Tangent Line

Documents

Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Monday, 08 June 2015Created by Mr Lafferty1 Circles Revision of Angle Properties Angles in