The Circle
S3
Credit
Isosceles Triangles in Circles
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Right angle in a Semi-Circle
Tangent Line to a Circle
Diameter Symmetry in a Circle
Circumference of a Circle
Length of an ARC of a Circle
Area of a Circle
Area of a SECTOR of a Circle
Summary of Circle Chapter
Wednesday, 23 November Created by Mr Lafferty 1
2011
Starter Questions
S3
Credit
Q1. True or false
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5( x 2) 2 x( x 3) 2 x 11x 10
2
Q2. How many degrees in one eighth of a circle.
Q3. Does x 2 8x 12 factorise to (x - 6)(x - 2)
Q4. After a discount of 20% an iPod is £160.
How much was it originally.
Wednesday, 23 November Created by Mr Lafferty 2
2011
Isosceles triangles
S3
Credit
in Circles
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Aim of Today’s Lesson
To identify isosceles triangles
within a circle.
Wednesday, 23 November Created by Mr Lafferty 3
2011
Isosceles triangles
S3
Credit
in Circles
When two radii are drawn to the ends of a chord,
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An isosceles triangle is formed.
A xo
B
xo
Online Demo
C
Wednesday, 23 November Created by Mr Lafferty 4
2011
Isosceles triangles
S3
Credit
in Circles
Special Properties of Isosceles Triangles
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Two equal lengths
Two equal angles
o
Angles in any triangle sum to 180
Wednesday, 23 November Created by Mr Lafferty 5
2011
Isosceles triangles
S3
Credit
in Circles
o
Q. Find the angle x .
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Solution
Angle at C is equal to:
B
360o 280o 80o
o
C
Since the triangle is isosceles A x o
we have 280
2x o 80o 180o
180o 80o 2x o
2x o 100o
x o 50o
Wednesday, 23 November Created by Mr Lafferty 6
2011
Isosceles triangles
S3
Credit
in Circles
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Maths in Action Ex 2.1 page 181
Wednesday, 23 November Created by Mr Lafferty 7
2011
Starter Questions
S3
Credit
Q1. Explain how we solve 5( x 2) 20
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Q2. How many degrees in one tenth of a circle.
Q3. Factorise x 2 13x 42
Q4. After a discount of 40% a Digital Radio is £120.
Explain why the originally price was £200.
Wednesday, 23 November Created by Mr Lafferty 8
2011
Semi-circle angle
S3
Credit
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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.
Wednesday, 23 November Created by Mr Lafferty 9
2011
Semi-circle angle
S3
Credit
Tool-kit required
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1. Protractor
2. Pencil
3. Ruler
Wednesday, 23 November Created by Mr Lafferty 10
2011
Semi-circle angle
S3
Credit
1. Using your pencil trace round
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the protractor so that you have
semi-circle.
2. Mark the centre of
the semi-circle.
You should have
something like this.
Wednesday, 23 November Created by Mr Lafferty 11
2011
Semi-circle angle
S3
Credit
x x x
Mark three points x
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1. Outside the circle x x x
2. On the circumference
x x
3. Inside the circle
Wednesday, 23 November Created by Mr Lafferty 12
2011
Semi-circle angle
S3
Credit
For each of the points
x
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x
Form a triangle by drawing a
line from each end of the
diameter to the point. x
Measure the angle at the
various points.
Log your results in a table.
Outside Circumference Inside
Wednesday, 23 November Created by Mr Lafferty 13
2011
Semi-circle angle
S3
Credit
Online Demo x
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x
x
Outside Circumference
Inside
o o o
90
Begin Maths in Action Book page 182
Wednesday, 23 November Created by Mr Lafferty 14
2011
Starter Questions
S3
Credit
If a = 7 b = 4 and c = 10
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Write down as many equations as you can
e.g. a + b = 11
Wednesday, 23 November Created by Mr Lafferty 15
2011
Tangent line
S3
Credit
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Aim of Today’s Lesson
To understand what a tangent line is
and its special property with the
radius at the point of contact.
Wednesday, 23 November Created by Mr Lafferty 16
2011
Tangent line
S3
Credit
A tangent line is a line that
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touches a circle at only one point.
Which of the
lines are
tangent to
the circle?
Wednesday, 23 November Created by Mr Lafferty 17
2011
Tangent line
S3
Credit
The radius of the circle that touches the tangent
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line is called the point of contact radius.
Online Demo
Special Property
The point of contact radius
is always perpendicular
(right-angled)
to the tangent line.
Wednesday, 23 November Created by Mr Lafferty 18
2011
Tangent line
S3
Credit
Q. Find the length of the tangent line between
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A and B.
Solution
Right-angled at A since B
AC is the radius at the point
of contact with the 10
Tangent.
By Pythagoras Theorem we have
A C
a b c
2 2 2
8
a 8 10
2 2 2
a 2 102 82
a 2 100 64 36
a 36 6
Wednesday, 23 November Created by Mr Lafferty 19
2011
Tangent line
S3
Credit
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Maths in Action Ex 4.1 page 185
Wednesday, 23 November Created by Mr Lafferty 20
2011
Starter Questions
S3
Credit
Q1. Using FOIL multiply out (x2 + 4x - 3)(x + 1)
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Using a multiplication table
expand out (x2 + 4x - 3)(x + 1)
Q2. Factorise 4x 2 9
Q3. I want to make 15% profit on a computer
I bought for £980. How much must I sell it for.
Wednesday, 23 November Created by Mr Lafferty 21
2011
Diameter symmetry
S3
Credit
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Aim of Today’s Lesson
To understand
some special properties
when a diameter bisects a chord.
Wednesday, 23 November Created by Mr Lafferty 22
2011
Diameter symmetry
S3
Credit
C
1. A line drawn through the centre of a circle
through the midpoint a chord will ALWAYS cut
the chord at right-angles
O
2. A line drawn through the centre of a circle
at right-angles to a chord will
A B ALWAYS bisect that chord.
3. A line bisecting a chord at right angles
D will ALWAYS pass through the centre of a circle.
Wednesday, 23 November Created by Mr Lafferty 23
2011
Diameter symmetry
S3
Credit
Q. Find the length of the chord A and B.
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Solution
Radius of the circle is 4 + 6 = 10. B
Since yellow line bisect AB and passes 10
through centre O, triangle is right-angle.
By Pythagoras Theorem we have
O
4 6
a b c
2 2 2
Since AB is bisected
a 6 10
2 2 2
The length of AB is
a 2 102 62 A
a 2 100 36 64 lengthAB 2 8 16
a 64 8
Wednesday, 23 November Created by Mr Lafferty 24
2011
Diameter symmetry
S3
Credit
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Maths in Action
Ex 5.1 & Ex 5.2 page 187
Wednesday, 23 November Created by Mr Lafferty 25
2011
Starter Questions
S3
Credit
Q1.Factorise x 2 81
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12 4 4
Q2.True or false
28 3 7 8m
Q3. Explain why
12m
the area of the triangle is 48m2
Q4. Solve x 2 2x 3
Wednesday, 23 November Created by Mr Lafferty 26
2011
Circumference
S3
Credit
of a circle
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Aim of Today’s Lesson
To be able to use the formula
for calculating
the circumference of a circle
Wednesday, 23 November Created by Mr Lafferty 27
2011
Circumference
S3
Credit
of a circle
Main parts of the circle
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radius
O
Diameter
Circumference D 2r
C D
Wednesday, 23 November Created by Mr Lafferty 28
2011
Circumference
S3
Credit
of a circle
Q. Find the circumference of the circle ?
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Solution
4cm
C D
C 8
C 25.12cm
Wednesday, 23 November Created by Mr Lafferty 29
2011
Circumference
S3
Credit
of a circle
Q. The circumference of the circle is 60cm ?
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Find the length of the diameter and radius.
Solution
C D D 2r
60 D
19 2r
60
D cm 19
r
2
D 19 cm r 9.5cm
Wednesday, 23 November Created by Mr Lafferty 30
2011
Circumference
S3
Credit
of a circle
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Now it’s your turn !
Maths in Action Ex 7.1 page 191
Wednesday, 23 November Created by Mr Lafferty 31
2011
Starter Questions
S3
Credit
Q1. True or false 5(2 x 1) 2(5 x 1)
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Q2. Using the balancing method rearrange into D =
C D
72
Q3. Simplify
360
Q4. Calculate 2 21
1
7 36
Wednesday, 23 November Created by Mr Lafferty 32
2011
length of the
S3
arc of a circle
Credit
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Aim of Today’s Lesson
To find and be able to use the formula
for calculating the length of an arc.
Wednesday, 23 November Created by Mr Lafferty 33
2011
Arc length of a circle
S3
Credit
Q. What is an arc ?
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Answer
A
An arc is a fraction
of the circumference.
minor arc
B
major arc
Wednesday, 23 November Created by Mr Lafferty 34
2011
Arc length of a circle
S3
Credit
Q. Find the circumference of the circle ?
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Solution
10cm
C D
C 20
C 62.8cm
Wednesday, 23 November Created by Mr Lafferty 35
2011
Arc length of a circle
S3
Credit
Q. Find the length of the minor arc XY below ?
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x Arc length Arc angle
connection =
y πD 360o
o 6 cm
45
45o
arc length o
( 12)
360
o
360
arc length 4.71cm
Wednesday, 23 November Created by Mr Lafferty 36
2011
Arc length of a circle
S3
Credit
Q. Find the length of the minor arc AB below ?
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Arc length Arc angle
connection =
A πD 360o
9 cm
60
o 60o
arc length o
( 18)
360
B arc length 9.42cm
Wednesday, 23 November Created by Mr Lafferty 37
2011
Arc length of a circle
S3
Credit
Q. Find the length of the major arc PQ below ?
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Arc length Arc angle
connection =
P πD 360o
10 m
260 100
o o 260o
arc length o
( 20)
360
Q arc length 45.38cm
Wednesday, 23 November Created by Mr Lafferty 38
2011
length of the
S3
arc of a circle
Credit
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Now it’s your turn !
Maths in Action Ex 8.1 page 193
Wednesday, 23 November Created by Mr Lafferty 39
2011
Starter Questions
S3
Credit
Q1. True or false
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4 x 9 (2 x 3)(2 x 3)
2
Q2. Expand out (x + 3)(x2 + 40 – 9)
Q3. Does x 2 4x 4 factorise (x - 2)(x - 2)
Q4. I want to make 30% profit on a DVD player I
bought for £80. How much must I sell it for.
Wednesday, 23 November Created by Mr Lafferty 40
2011
The Area of a circle
S3
Credit
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Aim of Today’s Lesson
To come up with and be able to use
the formula for calculating
the area of a circle
Wednesday, 23 November Created by Mr Lafferty 41
2011
The Area of a circle
S3
Credit
If we break the circle
into equal sectors
And lay them out side by side
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We get very close
to a rectangle.
8 1
7 2
6 3
5 4 2 4 6 8
1 3 5 7
Wednesday, 23 November Created by Mr Lafferty 42
2011
The Area of a circle
S3
Credit
If we cut the sectors
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2 4 6 8 Thinner and thinner then
1 3 5 7 we get closer and closer
to a rectangle. Hence we can
represent the area of a circle
thinner and thinner
by a rectangle.
sectors
r
r
Wednesday, 23 November Created by Mr Lafferty 43
2011
The Area of a circle
S3
Credit
r
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r
Area of a rectangle l b
Area of a rectangle r r r 2
But the area inside this rectangle is also the area of the circle
Area of a circle r 2
Wednesday, 23 November Created by Mr Lafferty 44
2011
The Area of a circle
S3
Credit
Q. Find the area of the circle ?
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Solution
4cm
A r 2
A 42
A 50.26cm 2
Wednesday, 23 November Created by Mr Lafferty 45
2011
The Area of a circle
S3
Credit
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Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q1-2
Wednesday, 23 November Created by Mr Lafferty 46
2011
The Area of a circle
S3
Credit
Q. The diameter of the circle is 60cm.
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Find area of the circle?
Solution
A r 2
60 D
r 30cm
2 2
A 302
A 2827.43cm 2
Wednesday, 23 November Created by Mr Lafferty 47
2011
The Area of a circle
S3
Credit
www.mathsrevision.com
Now it’s your turn !
Maths in Action
Ex9.1 page 194
Wednesday, 23 November Created by Mr Lafferty 48
2011
The Area of a circle
S3
Credit
2
Q. The area of a circle is 12.64 cm .
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Find its radius?
Solution
A r 2
12.64 r 2
12.64
r
2
4cm
r 4 2cm
Wednesday, 23 November Created by Mr Lafferty 49
2011
The Area of a circle
S3
Credit
www.mathsrevision.com
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q5 onwards
Wednesday, 23 November Created by Mr Lafferty 50
2011
Starter Questions
S3
Credit
Q1. Find the missing numbers
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4 x 12 x 9 (2 x ?)(? x ?)
2
Q2. Using the balancing method rearrange into x =
y 1 x
2 5
Q3. Calculate 1 2
5 8
Wednesday, 23 November Created by Mr Lafferty 51
2011
Sector area of a circle
S3
Credit
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Aim of Today’s Lesson
To find and be able to use the formula
for calculating the sector of an circle.
Wednesday, 23 November Created by Mr Lafferty 52
2011
Area of Sector in a circle
S3
Credit
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A
minor sector
B
major sector
Wednesday, 23 November Created by Mr Lafferty 53
2011
Area of Sector in a circle
S3
Credit
Q. Find the area of the circle ?
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Solution
10cm A r 2
A 102
A 314cm 2
Wednesday, 23 November Created by Mr Lafferty 54
2011
Area of Sector in a circle
S3
Credit
Find the area of the minor sector XY below ?
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x connection
Area Sector = Sector angle
y
πr2 360o
o 6 cm
45
45o
o
Area of Sector ( 62 )
360 360o
Area Sector 14.14cm 2
Wednesday, 23 November Created by Mr Lafferty 55
2011
Area of Sector in a circle
S3
Credit
Q. Find the area of the minor sector AB below ?
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connection
A Area Sector Sector angle
=
πr2 360o
9 cm
60
o 60o
Area Sector o
( 9 )
2
360
B
Area Sector 42.41cm 2
Wednesday, 23 November Created by Mr Lafferty 56
2011
Area of Sector in a circle
S3
Credit
Q. Find the area of the major sector PQ below ?
connection
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Sector Area Sector angle
=
P πr2 360o
10 m
260 100
o o 260o
Sector Area ( 102 )
360o
Q
Area Sector 226.89cm 2
Wednesday, 23 November Created by Mr Lafferty 57
2011
Sector area of a circle
S3
Credit
www.mathsrevision.com
Now it’s your turn !
Maths in Action Ex 10.1 page 196
Wednesday, 23 November Created by Mr Lafferty 58
2011
Starter Questions
S3
Credit
Q1. Using the balancing method rearrange into x =
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1
x 2 y 10
2
Q2. True or false 2(x - 3) + 3x = 5x - 6
Q3. Factorise 3x 2 x 2
Wednesday, 23 November Created by Mr Lafferty 59
2011
Summary of Circle Topic
line that bisects a chord
Arc length is 1. Splits the chord into 2
S3
equal halves.
Credit Arclength centre angleo 2. Makes right-angle with Pythagoras Theorem
= SOHCAHTOA
D 360o the chord.
3. Passes through centre
of the circle
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Circumference Semi-circle angle is
is o
C D always 90
Area is
Tangent touches
circle at one point
and make angle 90
o A r 2
with point of contact
radius
Diameter Radius
1
D 2r r D
2
Sector area
Areasector centre angleo
Wednesday, 23 November Created by Mr Lafferty = 60
2011 r 2 360o
Summary of Circle Topic
S3
Credit
www.mathsrevision.com
Maths in Action Book Page 199
Wednesday, 23 November Created by Mr Lafferty 61
2011