The Circle

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The Circle - PowerPoint PPT Presentation

The Circle. Length of Arc in a Circles . Area of a Sector in a Circle. Mix Problems including Angles. Symmetry in a Circle. Diameter & Right Angle in a Circle. Semi-Circle & Right Angle in a Circle. Tangent & Right Angle in a Circle. Summary of Circle Chapter. Starter Questions.

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The Circle

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The Circle

Length of Arc in a Circles

Area of a Sector in a Circle

Mix Problems including Angles

Symmetry in a Circle

Diameter & Right Angle in a Circle

Semi-Circle & Right Angle in a Circle

Tangent & Right Angle in a Circle

Summary of Circle Chapter

Starter Questions

Q1. Simplify

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.

Arc length of a circle

Aim of Today’s Lesson

To find and be able to use the formula

for calculating the length of an arc.

minor arc

major arc

Q. What is an arc ?

Arc length of a circle

An arc is a fraction

of the circumference.

A

B

Solution

Q. Find the circumference of the circle ?

Arc length of a circle

10cm

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

360o

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

60o

B

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

Arc length of a circle

Arc length

Arc angle

=

connection

A

πD

360o

10 m

260o

100o

B

Arc length of a circle

Now try Ex 1

Ch6 MIA (page 60)

Starter Questions

Q1. Solve

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

Q3.

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

How much was it originally.

Sector area of a circle

Aim of Today’s Lesson

To find and be able to use the formula

for calculating the sector of an circle.

minor sector

major sector

Area of Sector in a circle

A

B

Solution

Q. Find the area of the circle ?

Area of Sector in a circle

10cm

Find the area of the minor sector XY below ?

Area of Sector in a circle

x

connection

Area Sector

Sector angle

=

πr2

y

360o

6 cm

45o

360o

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

60o

B

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

Area of Sector in a circle

connection

Sector Area

Sector angle

=

A

πr2

360o

10 m

260o

100o

B

Area of Sector in a circle

Now try Ex 2

Ch6 MIA (page 62)

Starter Questions

Q1. Find the gradient and y - intercept

Q2. Expand out (x2 + 4x - 3)(x + 1)

Q3.

Q4. I want to make 15% profit on a computer I

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

Q. The arc length is 7.07cm. Find the angle xo

Arc length of a circle

Arc length

Arc angle

=

connection

A

πD

360o

9 cm

xo

B

Q. Find the angle given area of sector AB is 235.62cm2 ?

Area of Sector in a circle

connection

Sector Area

Sector angle

=

A

πr2

360o

10 m

xo

B

Mixed Problems

Now try Ex 3

Ch6 MIA (page 64)

Starter Questions

Q1. Find the gradient and y - intercept

Q2. Expand out (x + 3)(x2 + 40 – 9)

Q3.

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

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

Isosceles triangles in Circles

Aim of Today’s Lesson

To identify isosceles triangles

within a circle.

A

B

Isosceles triangles in Circles

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

An isosceles triangle is formed.

xo

xo

Online Demo

C

Isosceles triangles in Circles

Special Properties of Isosceles Triangles

Two equal lengths

Two equal angles

Angles in any triangle sum to 180o

Solution

Angle at C is equal to:

Isosceles triangles in Circles

Q. Find the angle xo.

B

C

xo

Since the triangle is isosceles

we have

A

280o

Isosceles triangles in Circles

Now try Ex 4

Ch6 MIA (page 66)

Starter Questions

Q1. Find the gradient and y - intercept

Q2. Expand out 2(x - 3) + 3x

Q3.

Q4. Car depreciates by 20% each year. How much is

it worth after 3 years.

Diameter symmetry

Aim of Today’s Lesson

To understand

some special properties

when a diameter bisects a chord.

C

A

B

D

Diameter symmetry

• A line drawn through the centre of a circle
• and 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 (cut equally in 2) that chord.
• A line bisecting a chord at right angles
• will ALWAYS pass through the centre of a circle.

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

By Pythagoras Theorem we have

4

6

Since AB is bisected

The length of AB is

A

A

B

Diameter symmetry

Find the length of OM and CM. Radius of circle is 5cm,

AB is 8cm and M is midpoint of AB.

C

Radius of the circle is 5cm.

AM is 8 ÷ 2 = 4

o

By Pythagoras Theorem we have

M

D

CM = 3 + 5 = 8cm

Symmetry & Chords in Circles

Now try Ex 14.2

Ch14 (page 188)

Starter Questions

Q1. Find the gradient and y - intercept

Q2. Expand out (x + 3)(x + 4)

Q3.

Q4. Bacteria increase at a rate of 10% per hour.

If there were 100 bacteria initially. How many are

there after 9 hours.

Semi-Circle Angle

Aim of Today’s Lesson

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.

Semi-circle angle

Tool-kit required

1. Protractor

2. Pencil

3. Ruler

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.

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

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

Outside

Circumference

Inside

Semi-circle angle

x

Online Demo

x

x

= 90o

> 90o

< 90o

Every angle in a semi-circle is a right angle

Angles in a Semi-Circle

Now try Ex 7

Ch6 MIA (page 71)

Starter Questions

Q1. Find the gradient and y - intercept

Q2. Expand out (a - 3)(a2 + 3a – 7)

Q3.

Q4. I want to make 60% profit on a TV I bought for

£240. How much must I sell it for.

Tangent line

Aim of Today’s Lesson

To understand what a tangent line is

and its special property with the

radius at the point of contact.

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.

Tangent line

The radius of the circle that touches the tangent

line is called the point of contact radius.

Online Demo

Special Property

is always perpendicular

(right-angled)

to the tangent line.

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

By Pythagoras Theorem we have

C

A

8

Tangent Lines

Now try Ex 8

Ch6 MIA (page 73)

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