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The Circle. Isosceles Triangles in Circles . Right angle in a Semi-Circle. Tangent Line to a Circle. Diameter Symmetry in a Circle. Circumference of a Circle. www.mathsrevision.com. Length of an ARC of a Circle. Area of a Circle. Area of a SECTOR of a Circle. Summary of Circle Chapter.

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The circle

The Circle

Isosceles Triangles in Circles

Right angle in a Semi-Circle

Tangent Line to a Circle

Diameter Symmetry in a Circle

Circumference of a Circle

www.mathsrevision.com

Length of an ARC of a Circle

Area of a Circle

Area of a SECTOR of a Circle

Summary of Circle Chapter

Created by Mr Lafferty


Starter questions

Starter Questions

Q1. True or false

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

Q3.

Q4. After a discount of 20% an iPod is £160.

How much was it originally.

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Isosceles Triangles in Circles

Aim of Today’s Lesson

We are learning to identify

isosceles triangles

within a circle.

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Isosceles triangles in Circles

When two radii are drawn to the ends of a chord,

an isosceles triangle is formed.

DEMO

A

B

xo

xo

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C

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Isosceles triangles in Circles

Special Properties of Isosceles Triangles

Two equal lengths

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Two equal angles

Angles in any triangle sum to 180o

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Solution

Angle at C is equal to:

Isosceles triangles in Circles

Q. Find the angle xo.

B

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C

xo

A

Since the triangle is isosceles

we have

280o

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Isosceles triangles in Circles

Maths in Action Ex 2.1 page 181

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Starter questions1

Starter Questions

Q1. Explain how we solve

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

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Q3.

Q4. After a discount of 40% a Digital Radio is £120.

Explain why the originally price was £200.

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Semi circle angle

Semi-circle angle

Aim of Today’s Lesson

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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Semi-circle angle

Tool-kit required

1. Protractor

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2. Pencil

3. Ruler

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You should have

something like this.

Semi-circle angle

1. Using your pencil trace round

the protractor so that you have

semi-circle.

2. Mark the centre of the semi-circle.

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Semi-circle angle

x

x

x

x

  • Mark three points

  • Outside the circle

x

x

x

2. On the circumference

x

x

3. Inside the circle

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Log your results in a table.

Outside

Circumference

Inside

Semi-circle angle

x

For each of the points

Form a triangle by drawing a

line from each end of the

diameter to the point.

Measure the angle at the

various points.

x

x

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DEMO

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Outside

Circumference

Inside

Semi-circle angle

DEMO

x

x

x

= 90o

> 90o

< 90o

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Begin Maths in Action Book page 182

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Starter questions2

Starter Questions

If a = 7 b = 4 and c = 10

Write down as many equations as you can

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e.g. a + b = 11

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Tangent line

Aim of Today’s Lesson

We are learning to understand

what a tangent line is and its

special property with the

radius at the point of contact.

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Which of the

lines are

tangent to

the circle?

Tangent line

A tangent line is a line that

touches a circle at only one point.

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Tangent line

The radius of the circle that touches the tangent

line is called the point of contact radius.

DEMO

Special Property

The point of contact radius

is always perpendicular

(right-angled)

to the tangent line.

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Solution

Right-angled at A since

AC is the radius at the point

of contact with the

Tangent.

Tangent line

Q. Find the length of the tangent line between

A and B.

B

10

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By Pythagoras Theorem we have

C

A

8

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Tangent line

Maths in Action Ex 4.1 page 185

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Starter questions3

Starter Questions

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

Using a multiplication table

expand out (x2 + 4x - 3)(x + 1)

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Q2.

Q3. I want to make 15% profit on a computer

I bought for £980. How much must I sell it for.

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Diameter symmetry

Aim of Today’s Lesson

We are learning to understand

some special properties

when a diameter bisects a chord.

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C

A

B

D

Diameter symmetry

  • A line drawn through the centre of a circle

  • through the midpoint a chord will ALWAYS cut

  • the chord at right-angles

O

  • A line drawn through the centre of a circle

  • at right-angles to a chord will

  • ALWAYS bisect that chord.

  • A line bisecting a chord at right angles

  • will ALWAYS pass through the centre of a circle.

DEMO

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Diameter symmetry

Q. Find the length of the chord A and B.

Solution

Radius of the circle is 4 + 6 = 10.

B

Since yellow line bisect AB and passes

through centre O, triangle is right-angle.

10

O

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By Pythagoras Theorem we have

4

6

Since AB is bisected

The length of AB is

A

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Diameter symmetry

Maths in Action

Ex 5.1 & Ex 5.2 page 187

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Starter questions4

8m

12m

Starter Questions

Q1.

Q2.

Q3. Explain why

the area of the triangle is 48m2

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Q4.

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Circumference of a circle

Circumference of a circle

Aim of Today’s Lesson

We are learning to use the formula

for calculating

the circumference of a circle

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Circumference of a circle1

radius

Diameter

Circumference

Circumference of a circle

Main parts of the circle

O

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Circumference of a circle2

Solution

Q.Find the circumference of the circle ?

Circumference of a circle

4cm

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Circumference of a circle3

Solution

  • The circumference of the circle is 60cm ?

  • Find the length of the diameter and radius.

Circumference of a circle

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Circumference of a circle4

Circumference of a circle

Now it’s your turn !

Maths in Action Ex 7.1 page 191

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Starter questions5

Starter Questions

Q1. True or false

Q2. Using the balancing method rearrange into D =

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Q3.

Q4. Calculate

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Length of the arc of a circle

length of the arc of a circle

Aim of Today’s Lesson

We are learning to use the formula

for calculating the length of an arc.

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Arc length of a circle

minor arc

major arc

Q. What is an arc ?

Arc length of a circle

Answer

An arc is a fraction

of the circumference.

A

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B

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Arc length of a circle1

Solution

Q. Find the circumference of the circle ?

Arc length of a circle

10cm

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Arc length of a circle2

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

Arc length of a circle

x

Arc length

Arc angle

=

connection

πD

360o

y

6 cm

45o

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360o

DEMO

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Arc length of a circle3

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

Arc length of a circle

Arc length

Arc angle

=

connection

A

πD

360o

9 cm

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60o

B

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Arc length of a circle4

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

Arc length of a circle

Arc length

Arc angle

=

connection

P

πD

360o

10 m

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260o

100o

Q

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Length of the arc of a circle1

length of the arc of a circle

Now it’s your turn !

Maths in Action Ex 8.1 page 193

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Starter questions6

Starter Questions

Q1. True or false

Q2. Expand out (x + 3)(x2 + 40 – 9)

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Q3.

Q4. I want to make 30% profit on a DVD player I

bought for £80. How much must I sell it for.

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The area of a circle

The Area of a circle

Aim of Today’s Lesson

We are learning to use

the formula for calculating

the area of a circle

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The area of a circle1

If we break the circle

into equal sectors

And lay them out side by side

We get very close

to a rectangle.

1

8

2

2

7

1

3

6

5

8

6

4

4

3

7

5

The Area of a circle

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The area of a circle2

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.

8

6

4

2

3

7

1

5

thinner and thinner

sectors

The Area of a circle

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The area of a circle3

The Area of a circle

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

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The area of a circle4

Solution

Q.Find the area of the circle ?

The Area of a circle

4cm

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The Area of a circle

Now it’s your turn !

Begin Maths in Action Book

Ex9.1 page 194 Q1-2

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The area of a circle5

Solution

  • The diameter of the circle is 60cm.

  • Find area of the circle?

The Area of a circle

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The Area of a circle

Now it’s your turn !

Maths in Action

Ex9.1 page 194

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The area of a circle6

Solution

  • The area of a circle is 12.64 cm2.

  • Find its radius?

The Area of a circle

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The Area of a circle

Now it’s your turn !

Begin Maths in Action Book

Ex9.1 page 194 Q5 onwards

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Starter questions7

Starter Questions

Q1. Find the missing numbers

Q2. Using the balancing method rearrange into x =

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Q3. Calculate

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Sector area of a circle

Sector area of a circle

Aim of Today’s Lesson

We are learning to use the formula

for calculating the sector of an circle.

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Area of sector in a circle

minor sector

major sector

Area of Sector in a circle

A

B

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Area of sector in a circle1

Solution

Q. Find the area of the circle ?

Area of Sector in a circle

10cm

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Area of sector in a circle2

Find the area of the minor sector XY below ?

Area of Sector in a circle

x

connection

Area Sector

Sector angle

=

y

πr2

360o

6 cm

45o

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360o

DEMO

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Area of sector in a circle3

Q. Find the area of the minor sector AB below ?

Area of Sector in a circle

connection

Area Sector

Sector angle

=

A

πr2

360o

9 cm

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60o

B

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Area of sector in a circle4

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

Area of Sector in a circle

connection

Sector Area

Sector angle

=

P

πr2

360o

10 m

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260o

100o

Q

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Sector area of a circle

Now it’s your turn !

Maths in Action Ex 10.1 page 196

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Starter questions8

Starter Questions

Q1. Using the balancing method rearrange into x =

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

Q3.

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Arc length is

Pythagoras Theorem

SOHCAHTOA

Circumference is

Area is

Radius

Diameter

Sector area

Summary of Circle Topic

  • line that bisects a chord

  • Splits the chord into 2 equal halves.

  • Makes right-angle with the chord.

  • Passes through centre of the circle

Semi-circle angle is

always 90o

Tangent touches circle at one point

and make angle 90o with point of contact radius

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Summary of Circle Topic

Maths in Action Book Page 199

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