The Circle

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# The Circle - PowerPoint PPT Presentation

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

Isosceles Triangles in Circles

Right angle in a Semi-Circle

Tangent Line to a Circle

Diameter Symmetry in a Circle

Circumference of a Circle

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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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### 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 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

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 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

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 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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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

Aim of Today’s Lesson

We are learning to use the formula

for calculating

the circumference of a circle

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Diameter

Circumference

### Circumference of a circle

Main parts of the circle

O

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Solution

Q.Find the circumference of the circle ?

### Circumference of a circle

4cm

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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 circle

Maths in Action Ex 7.1 page 191

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### 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

Aim of Today’s Lesson

We are learning to use the formula

for calculating the length of an arc.

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minor arc

major arc

Q. What is an arc ?

### Arc length of a circle

An arc is a fraction

of the circumference.

A

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B

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Solution

Q. Find the circumference of the circle ?

### Arc length of a circle

10cm

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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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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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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 circle

Maths in Action Ex 8.1 page 193

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### 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

Aim of Today’s Lesson

We are learning to use

the formula for calculating

the area of a circle

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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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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 circle

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

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Solution

Q.Find the area of the circle ?

### The Area of a circle

4cm

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The Area of a circle

Begin Maths in Action Book

Ex9.1 page 194 Q1-2

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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

Maths in Action

Ex9.1 page 194

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Solution

• The area of a circle is 12.64 cm2.

### The Area of a circle

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The Area of a circle

Begin Maths in Action Book

Ex9.1 page 194 Q5 onwards

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### 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

Aim of Today’s Lesson

We are learning to use the formula

for calculating the sector of an circle.

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minor sector

major sector

### Area of Sector in a circle

A

B

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Solution

Q. Find the area of the circle ?

### Area of Sector in a circle

10cm

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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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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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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

Maths in Action Ex 10.1 page 196

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### 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

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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