Transcript Arc length of a circle

The Circle
www.mathsrevision.com
Isosceles Triangles in Circles
Right angle in a Semi-Circle
Tangent Line to a Circle
The Tangent Kite
Finding an ARC length
Finding the area of a SECTOR
Mixed questions
Finding the centre angle given ARC length
Finding the centre angle given SECTOR area
Exam questions
Friday, 17 July 2015
Created by Mr Lafferty
1
Starter Questions
www.mathsrevision.com
Level 4
Q1. True or false
5( x  2)  2x( x  3)  2x  11x  10
2
Q2. How many degrees in one eighth of a circle.
Q3. After a discount of 20% an iPod is £160.
How much was it originally.
Friday, 17 July 2015
Created by Mr Lafferty
2
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
Aim of Today’s Lesson
To identify isosceles triangles
within a circle.
Friday, 17 July 2015
Created by Mr Lafferty
3
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
When two radii are drawn to the ends of a chord,
An isosceles triangle is formed.
DEMO
A
xo
xo
B
C
Friday, 17 July 2015
Created by Mr Lafferty
4
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180
Friday, 17 July 2015
Created by Mr Lafferty
o
5
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
Q.
o
Find the angle x .
Solution
Angle at C is equal to:
B
360o  280o  80o
o
Since the triangle is isosceles
A x
we have2 xo  80o  180o
C
o
280
2 xo  100o
xo  50o
Friday, 17 July 2015
Created by Mr Lafferty
6
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180
Friday, 17 July 2015
Created by Mr Lafferty
o
7
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
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
10
through centre O, triangle is right-angle.
O
By Pythagoras Theorem we have
6
4
2
2
2
a b c
a 2  62  102
a 2  102  62
Since AB is bisected
The length of AB is
a 2  100  36  64
a  64  8
Friday, 17 July 2015
A
lengthAB  2  8  16
Created by Mr Lafferty
8
Isosceles triangles
in Circles
www.mathsrevision.com
Level 4
Now try N4+ TJ
Ex18.1
Ch18 (page 139)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
Level 4
Q1. Explain how we solve
5( x  2)  20
Q2. How many degrees in one tenth of a circle.
Q3. After a discount of 40% a Digital Radio is £120.
Explain why the originally price was £200.
Friday, 17 July 2015
Created by Mr Lafferty
10
Semi-circle angle
www.mathsrevision.com
Level 4
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.
Friday, 17 July 2015
Created by Mr Lafferty
11
Semi-circle angle
www.mathsrevision.com
National 5
Tool-kit required
1.
Protractor
2.
Pencil
3.
Ruler
Friday, 17 July 2015
Created by Mr Lafferty
12
Semi-circle angle
www.mathsrevision.com
Level 4
1.
Using your pencil trace round
the protractor so that you have
semi-circle.
2.
Mark the centre of
the semi-circle.
You should have
something like this.
Friday, 17 July 2015
Created by Mr Lafferty
13
Semi-Circle Angle
www.mathsrevision.com
Level 4
x
Mark three points
1. Outside the circle
2. On the circumference
3. Inside the circle
Friday, 17 July 2015
x
x
x
x
x
x
Created by Mr Lafferty
x
x
14
Semi-Circle Angle
www.mathsrevision.com
Level 4
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
DEMO
Log your results in a table.
Outside
Friday, 17 July 2015
Circumference
Created by Mr Lafferty
Inside
15
Semi-Circle Angle
www.mathsrevision.com
Level 4
DEMO
x
x
Outside
< 90
Circumference
o
= 90
o
x
Inside
> 90
o
Begin Maths in Action Book page 182
Friday, 17 July 2015
Created by Mr Lafferty
16
Semi-Circle Angle
www.mathsrevision.com
Level 4
Now try N4+ TJ
Ex18.2
Ch18 (page 141)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
Level 4
If a = 7 b = 4 and c = 10
Write down as many equations as you can
e.g. a + b = 11
Friday, 17 July 2015
Created by Mr Lafferty
18
Tangent line
www.mathsrevision.com
Level 4
Aim of Today’s Lesson
To understand what a tangent line is
and its special property with the
radius at the point of contact.
Friday, 17 July 2015
Created by Mr Lafferty
19
Tangent line
www.mathsrevision.com
Level 4
A tangent line is a line that
touches a circle at only one point.
Which of the
lines are
tangent to
the circle?
Friday, 17 July 2015
Created by Mr Lafferty
20
Tangent line
www.mathsrevision.com
Level 4
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.
Friday, 17 July 2015
Created by Mr Lafferty
21
Tangent line
www.mathsrevision.com
Level 4
Q.
Find the length of the tangent line between
A and B.
Solution
B
Right-angled at A since
AC is the radius at the point
10
of contact with the Tangent.
By Pythagoras Theorem we have A
C
8
a 2  b2  c 2
a 2  82  102
a 2  102  82
a 2  100  64  36
Friday, 17 July 2015
a  36  6
Created by Mr Lafferty
24
Tangent Line
www.mathsrevision.com
Level 4
Now try N4+ TJ
Ex18.3
Ch18.3 (page 143)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Q1. True or false
6 x  8x  2 x(4 x  4)
2
Q2. Expand out (x + 3)(x – 2)
Friday, 17 July 2015
Created by Mr Lafferty
27
Tangent Kite
www.mathsrevision.com
National 5
Learning Intention
We are learning properties of a
tangent kite.
Success Criteria
1. Know the properties of
tangent kites.
2. Be able to calculate lengths
and angles related to a
tangent kite.
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Tangent Kite
A tangent kite has two right-angles
A tangent kite is made up of two congruent triangles
17-Jul-15
Compiled by Mr. Lafferty Maths
Dept.
Tangent Kite
A
90o
138o
42o
90o
B
Find all angles in the tangent kite.
17-Jul-15
Compiled by Mr. Lafferty Maths
Dept.
C
Tangent Kite
r
Using Pythagoras Theorem
r2 = 502 - 402
r2 = 2500 - 1600
√
17-Jul-15
r2 = 900
r = 30cm
A = πr2
A = π(30)2
A = 2827cm2
Compiled by Mr. Lafferty Maths
Dept.
50cm
Find the area of the circle.
40cm
Tangent Line
www.mathsrevision.com
Level 4
Now try N4+ TJ
Ex18.4
Ch18.3 (page 145)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Q1. True or false
4 x  2 x  2 x(2 x 1)
2
Q2. Expand out (x + 3)(x2 + 40 – 9)
Q3. I want to make 30% profit on a DVD player I
bought for £80. How much must I sell it for.
Friday, 17 July 2015
Created by Mr Lafferty
33
Arc length of a circle
www.mathsrevision.com
National 5
Learning Intention
We are learning to use the arc
length formula.
Success Criteria
1. Know the arc length formula.
2. Be able to calculate arc
length showing appropriate
work.
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Arc length of a circle
www.mathsrevision.com
National 5
Q.
What is an arc ?
Answer
A
An arc is a fraction
of the circumference.
minor arc
B
major arc
Friday, 17 July 2015
Created by Mr Lafferty
35
Arc length of a circle
www.mathsrevision.com
National 5
Q.
Find the circumference of the circle ?
Solution
10cm
C  D
C    20
C  62.8cm
Friday, 17 July 2015
Created by Mr Lafferty
36
Arc length of a circle
www.mathsrevision.com
National 5
Q.
Find the length of the minor arc XY below ?
x
y
45
o
6 cm
360
Arc length Arc angle
=
connection
πD
360o
45o
arc length 
 (  12)
o
360
o
DEMO
Friday, 17 July 2015
arc length  4.71cm
Created by Mr Lafferty
37
Arc length of a circle
www.mathsrevision.com
National 5
Q.
Find the length of the minor arc AB below ?
Arc length Arc angle
=
connection
A
πD
360o
9 cm
60
60o
arc length 
 (  18)
o
360
o
B
Friday, 17 July 2015
arc length  9.42cm
Created by Mr Lafferty
38
Arc length of a circle
www.mathsrevision.com
National 5
Q.
Find the length of the major arc PQ below ?
Arc length Arc angle
=
connection
P
πD
360o
10 m
o
260o
arc length 
 (  20)
o
360
o
260 100
Q
Friday, 17 July 2015
arc length  45.38cm
Created by Mr Lafferty
39
Arc length of a circle
www.mathsrevision.com
Nat 5
Now try N5 TJ
Ex13.1
Ch13 (page 126)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Q1. True or false
4 x  9  (2 x  3)(2 x  3)
2
Q2. Expand out (x + 3)(x2 + 40 – 9)
Q3. I want to make 30% profit on a DVD player I
bought for £80. How much must I sell it for.
Friday, 17 July 2015
Created by Mr Lafferty
41
Sector area of a circle
www.mathsrevision.com
Nat 5
Learning Intention
Success Criteria
We are learning how to find the
area of a sector.
1. Know the sector formula.
17-Jul-15
2. Be able to calculate sector
area showing appropriate
work.
Compiled by Mr. Lafferty Maths
Dept.
Area of Sector in a circle
www.mathsrevision.com
National 5
A
B
minor sector
major sector
Friday, 17 July 2015
Created by Mr Lafferty
43
Area of Sector in a circle
www.mathsrevision.com
National 5
Q.
Find the area of the circle ?
Solution
10cm
A  r 2
A    102
A  314cm
Friday, 17 July 2015
Created by Mr Lafferty
2
44
Area of Sector in a circle
www.mathsrevision.com
National 5
Find the area of the minor sector XY below ?
x
connection
Area Sector = Sector angle
y
2
πr
360o
o 6 cm
45
360
45o
2
Area of Sector 

(


6
)
o
360
o
DEMO
Friday, 17 July 2015
Area Sector  14.14cm 2
Created by Mr Lafferty
45
Area of Sector in a circle
www.mathsrevision.com
National 5
Q.
Find the area of the minor sector AB below ?
connection
A
9 cm
60
60o
2
Area Sector 

(


9
)
o
360
o
B
Friday, 17 July 2015
Area Sector
Sector angle
=
πr2
360o
Area Sector  42.41cm 2
Created by Mr Lafferty
46
Area of Sector in a circle
www.mathsrevision.com
National 5
Q.
Find the area of the major sector PQ below ?
connection
Sector Area
Sector angle
=
P
πr2
360o
10 m
o
260o
2
Sector Area 

(


10
)
o
360
o
260 100
Q
Friday, 17 July 2015
Area Sector  226.89cm 2
Created by Mr Lafferty
47
Arc length of a circle
www.mathsrevision.com
Nat 5
Now try N5 TJ
Ex13.2
Ch13 (page 127)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Q1. Using the balancing method rearrange into x =
1
x  2 y  10
2
Q2. True or false 2(x - 3) + 3x = 5x - 6
Q3.
Factorise 3x 2  9x
Friday, 17 July 2015
Created by Mr Lafferty
49
Mixed Question
www.mathsrevision.com
Nat 5
Learning Intention
We are doing mixed questions on
the circle.
17-Jul-15
Success Criteria
1. Identify appropriate formula
to use.
2. Be able to calculate sector /
arc showing appropriate
work.
Compiled by Mr. Lafferty Maths
Dept.
Arc length of a circle
www.mathsrevision.com
Nat 5
Now try N5 TJ
Ex13.3
Ch13 (page 128)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Rearrange the formula below into xo =
Al
xo

 D 360o
Friday, 17 July 2015
Created by Mr Lafferty
56
Centre Angle (Arc length)
www.mathsrevision.com
Nat 5
Learning Intention
We are learning how to find the
centre angle given the arc length.
17-Jul-15
Success Criteria
1. Know formula.
2. Be able to calculate centre
angle showing appropriate
work.
Compiled by Mr. Lafferty Maths
Dept.
Centre Angle (Arc length)
www.mathsrevision.com
National 5
Find the centre angle xo given the arc length is 4.71cm2.
x
y
45
o
6 cm
Arc length Arc angle
=
connection
πD
360o
4.71
x 
 360o
  12
o
xo  45o
Friday, 17 July 2015
Created by Mr Lafferty
58
Centre Angle (Arc length)
www.mathsrevision.com
Nat 5
Now try N5 TJ
Ex13.4
Ch13 (page 129)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
Starter Questions
www.mathsrevision.com
National 5
Rearrange the formula below into xo =
AoS
2
r
Friday, 17 July 2015

xo
o
360
Created by Mr Lafferty
61
Centre Angle (AoS)
www.mathsrevision.com
Nat 5
Learning Intention
We are learning how to find the
centre angle given the sector area.
17-Jul-15
Success Criteria
1. Know formula.
2. Be able to calculate centre
angle showing appropriate
work.
Compiled by Mr. Lafferty Maths
Dept.
Centre Angle (AoS)
www.mathsrevision.com
National 5
Find the angle xo given that the sector area is 42.41cm2.
connection
A
9 cm
x
Area Sector
Sector angle
=
πr2
360o
42.41
o
x 

360
  92
o
o
B
Friday, 17 July 2015
xo  60o
Created by Mr Lafferty
63
Centre Angle (AoS)
www.mathsrevision.com
National 5
Find the angle xo given that the sector area is 51.3mm2.
connection
A
7 mm
x
Area Sector
Sector angle
=
πr2
360o
51.3
o
x 

360
  72
o
o
B
Friday, 17 July 2015
xo  120o
Created by Mr Lafferty
64
Centre Angle (AoS)
www.mathsrevision.com
Nat 5
Now try N5 TJ
Ex13.5
Ch13 (page 130)
17-Jul-15
Compiled by Mr. Lafferty Maths Dept.
3 marks