the circle
Download
Skip this Video
Download Presentation
The Circle

Loading in 2 Seconds...

play fullscreen
1 / 61

The Circle - PowerPoint PPT Presentation


  • 103 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The Circle' - vita


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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

www.mathsrevision.com

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.

Created by Mr Lafferty

slide3

Isosceles Triangles in Circles

Aim of Today’s Lesson

We are learning to identify

isosceles triangles

within a circle.

www.mathsrevision.com

Created by Mr Lafferty

slide4

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

www.mathsrevision.com

C

Created by Mr Lafferty

slide5

Isosceles triangles in Circles

Special Properties of Isosceles Triangles

Two equal lengths

www.mathsrevision.com

Two equal angles

Angles in any triangle sum to 180o

Created by Mr Lafferty

slide6

Solution

Angle at C is equal to:

Isosceles triangles in Circles

Q. Find the angle xo.

B

www.mathsrevision.com

C

xo

A

Since the triangle is isosceles

we have

280o

Created by Mr Lafferty

slide7

Isosceles triangles in Circles

Maths in Action Ex 2.1 page 181

www.mathsrevision.com

Created by Mr Lafferty

starter questions1

Starter Questions

Q1. Explain how we solve

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

www.mathsrevision.com

Q3.

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

Explain why the originally price was £200.

Created by Mr Lafferty

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.

www.mathsrevision.com

Created by Mr Lafferty

slide10

Semi-circle angle

Tool-kit required

1. Protractor

www.mathsrevision.com

2. Pencil

3. Ruler

Created by Mr Lafferty

slide11

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.

www.mathsrevision.com

Created by Mr Lafferty

slide12

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

www.mathsrevision.com

Created by Mr Lafferty

slide13

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

www.mathsrevision.com

DEMO

Created by Mr Lafferty

slide14

Outside

Circumference

Inside

Semi-circle angle

DEMO

x

x

x

= 90o

> 90o

< 90o

www.mathsrevision.com

Begin Maths in Action Book page 182

Created by Mr Lafferty

starter questions2

Starter Questions

If a = 7 b = 4 and c = 10

Write down as many equations as you can

www.mathsrevision.com

e.g. a + b = 11

Created by Mr Lafferty

slide16

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.

www.mathsrevision.com

Created by Mr Lafferty

slide17

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.

www.mathsrevision.com

Created by Mr Lafferty

slide18

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.

www.mathsrevision.com

Created by Mr Lafferty

slide19

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

www.mathsrevision.com

By Pythagoras Theorem we have

C

A

8

Created by Mr Lafferty

slide20

Tangent line

Maths in Action Ex 4.1 page 185

www.mathsrevision.com

Created by Mr Lafferty

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)

www.mathsrevision.com

Q2.

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

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

Created by Mr Lafferty

slide22

Diameter symmetry

Aim of Today’s Lesson

We are learning to understand

some special properties

when a diameter bisects a chord.

www.mathsrevision.com

Created by Mr Lafferty

slide23

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

Created by Mr Lafferty

slide24

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

www.mathsrevision.com

By Pythagoras Theorem we have

4

6

Since AB is bisected

The length of AB is

A

Created by Mr Lafferty

slide25

Diameter symmetry

Maths in Action

Ex 5.1 & Ex 5.2 page 187

www.mathsrevision.com

Created by Mr Lafferty

starter questions4

8m

12m

Starter Questions

Q1.

Q2.

Q3. Explain why

the area of the triangle is 48m2

www.mathsrevision.com

Q4.

Created by Mr Lafferty

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

www.mathsrevision.com

Created by Mr Lafferty

circumference of a circle1

radius

Diameter

Circumference

Circumference of a circle

Main parts of the circle

O

www.mathsrevision.com

Created by Mr Lafferty

circumference of a circle2

Solution

Q.Find the circumference of the circle ?

Circumference of a circle

4cm

www.mathsrevision.com

Created by Mr Lafferty

circumference of a circle3

Solution

  • The circumference of the circle is 60cm ?
  • Find the length of the diameter and radius.

Circumference of a circle

www.mathsrevision.com

Created by Mr Lafferty

circumference of a circle4

Circumference of a circle

Now it’s your turn !

Maths in Action Ex 7.1 page 191

www.mathsrevision.com

Created by Mr Lafferty

starter questions5

Starter Questions

Q1. True or false

Q2. Using the balancing method rearrange into D =

www.mathsrevision.com

Q3.

Q4. Calculate

Created by Mr Lafferty

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.

www.mathsrevision.com

Created by Mr Lafferty

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

www.mathsrevision.com

B

Created by Mr Lafferty

arc length of a circle1

Solution

Q. Find the circumference of the circle ?

Arc length of a circle

10cm

www.mathsrevision.com

Created by Mr Lafferty

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

www.mathsrevision.com

360o

DEMO

Created by Mr Lafferty

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

www.mathsrevision.com

60o

B

Created by Mr Lafferty

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

www.mathsrevision.com

260o

100o

Q

Created by Mr Lafferty

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

www.mathsrevision.com

Created by Mr Lafferty

starter questions6

Starter Questions

Q1. True or false

Q2. Expand out (x + 3)(x2 + 40 – 9)

www.mathsrevision.com

Q3.

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

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

Created by Mr Lafferty

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

www.mathsrevision.com

Created by Mr Lafferty

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

www.mathsrevision.com

Created by Mr Lafferty

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

www.mathsrevision.com

Created by Mr Lafferty

the area of a circle3

The Area of a circle

www.mathsrevision.com

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

Created by Mr Lafferty

the area of a circle4

Solution

Q.Find the area of the circle ?

The Area of a circle

4cm

www.mathsrevision.com

Created by Mr Lafferty

slide46

The Area of a circle

Now it’s your turn !

Begin Maths in Action Book

Ex9.1 page 194 Q1-2

www.mathsrevision.com

Created by Mr Lafferty

the area of a circle5

Solution

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

The Area of a circle

www.mathsrevision.com

Created by Mr Lafferty

slide48

The Area of a circle

Now it’s your turn !

Maths in Action

Ex9.1 page 194

www.mathsrevision.com

Created by Mr Lafferty

the area of a circle6

Solution

  • The area of a circle is 12.64 cm2.
  • Find its radius?

The Area of a circle

www.mathsrevision.com

Created by Mr Lafferty

slide50

The Area of a circle

Now it’s your turn !

Begin Maths in Action Book

Ex9.1 page 194 Q5 onwards

www.mathsrevision.com

Created by Mr Lafferty

starter questions7

Starter Questions

Q1. Find the missing numbers

Q2. Using the balancing method rearrange into x =

www.mathsrevision.com

Q3. Calculate

Created by Mr Lafferty

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.

www.mathsrevision.com

Created by Mr Lafferty

area of sector in a circle

minor sector

major sector

Area of Sector in a circle

A

B

www.mathsrevision.com

Created by Mr Lafferty

area of sector in a circle1

Solution

Q. Find the area of the circle ?

Area of Sector in a circle

10cm

www.mathsrevision.com

Created by Mr Lafferty

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

www.mathsrevision.com

360o

DEMO

Created by Mr Lafferty

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

www.mathsrevision.com

60o

B

Created by Mr Lafferty

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

www.mathsrevision.com

260o

100o

Q

Created by Mr Lafferty

slide58

Sector area of a circle

Now it’s your turn !

Maths in Action Ex 10.1 page 196

www.mathsrevision.com

Created by Mr Lafferty

starter questions8

Starter Questions

Q1. Using the balancing method rearrange into x =

www.mathsrevision.com

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

Q3.

Created by Mr Lafferty

slide60

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

www.mathsrevision.com

Created by Mr Lafferty

slide61

Summary of Circle Topic

Maths in Action Book Page 199

www.mathsrevision.com

Created by Mr Lafferty

ad