Post on 08-Jan-2016
description
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 11
The CircleThe Circlew
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Finding an ARC length
Finding the area of a SECTOR
Finding the centre angle given ARC length
Finding the centre angle given SECTOR area
Mixed questions
Exam questions
Isosceles Triangles in Circles
Right angle in a Semi-Circle
Tangent Line to a Circle
The Tangent Kite
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 22
Starter QuestionsStarter Questions
Q1. True or false
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.
25( 2) 2 ( 3) 2 11 10 x x x x x
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Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 33
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To identify isosceles triangleswithin a circle.
Aim of Today’s Lesson
Isosceles triangles Isosceles triangles in Circlesin Circles
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 44
When two radii are drawn to the ends of a chord, When two radii are drawn to the ends of a chord, An isosceles triangle is formed.An isosceles triangle is formed.
C
A B
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
xo xoDemo
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 55
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180o
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
Level 4
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Q.Q.Find the angle xFind the angle xoo..
A
B
C
SolutionAngle at C is equal to:
Since the triangle is isosceleswe have
360 280 80o o o
xo
280o
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
2 80 180o o ox
2 100o ox
50o ox
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 77
Special Properties of Isosceles Triangles
Two equal lengths
Two equal angles
Angles in any triangle sum to 180o
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 88
Q.Q.Find the length of the chord A and B.Find the length of the chord A and B.
A
B
6O
Solution
By Pythagoras Theorem we have 2 2 2a b c
Isosceles triangles Isosceles triangles in Circlesin Circles
4
Since yellow line bisect AB and passesthrough centre O, triangle is right-angle.
Radius of the circle is 4 + 6 = 10.10
Since AB is bisected The length of AB is
2 8 16ABlength ww
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2 2 26 10a
2 2 210 6a
2 100 36 64a
64 8a
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N4+ TJEx18.1
Ch18 (page 139)
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Isosceles triangles Isosceles triangles in Circlesin Circles
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1010
Starter QuestionsStarter Questions
Q1. Explain how we solve
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.
5( 2) 20x
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Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1111
Semi-circle angleSemi-circle angle
To find the angle in a semi-circle made by a triangle with hypotenuse
equal to the diameter and the two smallerlengths meeting at the circumference.
Aim of Today’s Lesson
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National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1212
Semi-circle angleSemi-circle angle
Tool-kit requiredTool-kit required
1.1. ProtractorProtractor
2.2. PencilPencil
3.3. RulerRuler
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Level 4
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1.1. Using your pencil trace roundUsing your pencil trace roundthe protractor so that you havethe protractor so that you have semi-circle.semi-circle.
2.2. Mark the Mark the centre of centre of the the semi-circle.semi-circle.
You should have You should have something like this.something like this.
Semi-circle angleSemi-circle anglew
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Level 4
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Mark three points Mark three points
1.1. Outside the circleOutside the circle
x xx
x x
x x
x
x2. On the 2. On the
circumferencecircumference3. Inside the circle3. Inside the circle
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Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1515
For each of the points For each of the points
Form a triangle by drawing aForm a triangle by drawing aline from each end of the line from each end of the diameter to the point.diameter to the point.Measure the angle at the Measure the angle at the various points.various points.
x
x
x
Log your results in a table. Log your results in a table. InsideCircumferenc
eOutside
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Demo
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1616
InsideCircumferenceOutside
xx
x
< 90o
> 90o
= 90= 90oo
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mSemi-Circle AngleSemi-Circle Angle
Demo
Circle Circumference Angle - Investigation Worksheet
Outer Circumference
Inner
Circle 1
Circle 2
Circle 3
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N4+ TJEx18.2
Ch18 (page 141)
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Semi-Circle AngleSemi-Circle Angle
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1919
Starter QuestionsStarter Questionsw
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Write down as many equations as you can
e.g. a + b = 11
Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2020
To understand what a tangent line isand its special property with the
radius at the point of contact.
Aim of Today’s Lesson
Tangent lineTangent linew
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Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2121
Tangent lineTangent line
A A tangent line tangent line is a line that is a line that touches a circle at touches a circle at only one point.only one point.
Which of theWhich of thelines are lines are tangent to tangent to the circle?the circle?
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Level 4
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2222
Tangent lineTangent line
The radius of the circle that touches the tangent The radius of the circle that touches the tangent line is called line is called the point of contact radius.the point of contact radius.
Special PropertySpecial Property
The point of contact radiusThe point of contact radiusis always perpendicular is always perpendicular
(right-angled)(right-angled)to the tangent line.to the tangent line.
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Demo
Level 4
Right-angled at A sinceAC is the radius at the pointof contact with the Tangent.
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2525
Tangent lineTangent line
Q.Q.Find the length of the tangent line betweenFind the length of the tangent line between A and B.A and B.
A
B
8
10
C
Solution
By Pythagoras Theorem we have 2 2 2a b c
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2 2 28 10a
2 2 210 8a 2 100 64 36a 36 6a
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N4+ TJEx18.3
Ch18.3 (page 143)
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Tangent LineTangent Line
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2828
Starter QuestionsStarter Questions
Q1. True or false
Q2. Expand out (x + 3)(x – 2)
26 8 2 (4 4) x x x x
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National 5
Tangent KiteTangent Kite
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Know the properties of Know the properties of tangent kites.tangent kites.
We are learning properties of a tangent kite.
2.2. Be able to calculate Be able to calculate lengths and angles related lengths and angles related to a tangent kite.to a tangent kite.
20 Apr 2023 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
20 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Tangent Kite
Find all angles in the tangent kite.
42o
90o
90o
138o
A
B
C
20 Apr 2023 Compiled by Mr. Lafferty Maths Dept.
Tangent KiteFind the area of the circle.
40cm50
cm
Using Pythagoras Theorem
r
r2 = 502 - 402
r2 = 2500 - 1600
r2 = 900
r = 30cm√
A = πr2
A = π(30)2
A = 2827cm2
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N4+ TJEx18.4
Ch18.3 (page 145)
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Tangent LineTangent Line
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3434
Starter QuestionsStarter Questions
Q1. True or false
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.
24 2 2 (2 1) x x x x
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National 5
Arc length of a circleArc length of a circle
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Know the arc length Know the arc length formula.formula.
We are learning to use the arc length formula.
2.2. Be able to calculate arc Be able to calculate arc length showing length showing appropriate work.appropriate work.
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3636
Q. What is an arc ?
A
B
AnswerAnswer
An arc is a fractionof the circumference.
minor arc
major arcww
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circlecircle
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3737
Q. Find the circumference of the circle ?
SolutionSolution
C D 20C
62.8C cm
10cm
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mArc length of a Arc length of a
circlecircle
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3838
45
( 12)360
o
oarc length
length 4.71arc cm
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mArc length of a Arc length of a
circlecircle
Arc lengthπD
Arc angle
360o=
Q. Find the length of the minor arc XY below ?
6 cm45
o
x
y
360o
connection
DEMO
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3939
60
( 18)360
o
oarc length
length 9.42arc cm
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mArc length of a Arc length of a
circlecircle
Arc lengthπD
Arc angle
360o=
Q. Find the length of the minor arc AB below ?
9 cm
60o
A
B
connection
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4040
260
( 20)360
o
oarc length
length 45.38arc cm
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mArc length of a Arc length of a
circlecircle
Arc lengthπD
Arc angle
360o=
Q. Find the length of the major arc PQ below ?
10 m
100o
P
Q
connection
260o
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx13.1
Ch13 (page 126)
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Nat 5
Arc length of a Arc length of a circlecircle
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4242
Starter QuestionsStarter Questions
Q1. True or false
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.
24 9 (2 3)(2 3) x x x
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Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Sector area of a circleSector area of a circlew
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1.1. Know the sector formula.Know the sector formula.We are learning how to find the area of a sector.
2.2. Be able to calculate sector Be able to calculate sector area showing appropriate area showing appropriate work.work.
Nat 5
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4444
A
B
major sector
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mArea of Sector in a Area of Sector in a
circlecircle
minor sector
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4545
Q. Find the area of the circle ?
SolutionSolution
2A r 210A
2314A cm
10cm
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mArea of Sector in a Area of Sector in a
circlecircle
National 5
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245Area of Sector ( 6 )
360
o
o
2Area Sector 14.14 cm
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Area Sectorπr2
Sector angle
360o=
Find the area of the minor sector XY below ?
6 cm45
o
x
y
360o
connection
Area of Sector in a Area of Sector in a circlecircle
DEMO
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4747
260Area Sector ( 9 )
360
o
o
2 Sector 42.41Area cm
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Area Sectorπr2
Sector angle
360o=
Q. Find the area of the minor sector AB below ?
9 cm
A
B
connection
Area of Sector in a Area of Sector in a circlecircle
60o
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4848
2260Sector Area ( 10 )
360
o
o
2Area Sector 226.89 cm
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mArea of Sector in a Area of Sector in a
circlecircle
Sector Areaπr2
Sector angle360o
=
Q. Find the area of the major sector PQ below ?
10 m
100o
P
Q
connection
260o
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx13.2
Ch13 (page 127)
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Nat 5
Arc length of a Arc length of a circlecircle
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5050
Starter QuestionsStarter Questions
Q3.
Q1. Using the balancing method rearrange into x =
2Factorise 3 9x x
Q2. True or false 2(x - 3) + 3x = 5x - 6
12 10
2x y
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Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Mixed QuestionMixed Questionw
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1.1. Identify appropriate Identify appropriate formula to use.formula to use.
We are doing mixed questions on the circle.
2.2. Be able to calculate sector Be able to calculate sector / arc showing appropriate / arc showing appropriate work.work.
Nat 5
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx13.3
Ch13 (page 128)
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Nat 5
Arc length of a Arc length of a circlecircle
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5757
Starter QuestionsStarter Questions
Rearrange the formula below into xo =
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o
o
Al xD 360
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.ww
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1.1. Know formula.Know formula.We are learning how to find the centre angle given the arc length.
2.2. Be able to calculate centre Be able to calculate centre angle showing appropriate angle showing appropriate work.work.
Nat 5
Centre Angle (Arc Centre Angle (Arc length)length)
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5959
o o4.71x 360
12
o ox 45
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Arc lengthπD
Arc angle
360o=
Find the centre angle xo given the arc length is 4.71cm2.
6 cm45
o
x
yconnection
Centre Angle (Arc Centre Angle (Arc length)length)
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx13.4
Ch13 (page 129)
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Nat 5
Centre Angle (Arc Centre Angle (Arc length)length)
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6262
Starter QuestionsStarter Questions
Rearrange the formula below into xo =
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o o
2 o
A S x
r 360
Level 4
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Centre Angle (ACentre Angle (AooS)S)w
ww
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1.1. Know formula.Know formula.We are learning how to find the centre angle given the sector area.
2.2. Be able to calculate centre Be able to calculate centre angle showing appropriate angle showing appropriate work.work.
Nat 5
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6464
o o2
42.41x 360
9
o ox 60
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Area Sectorπr2
Sector angle
360o=
Find the angle xo given that the sector area is 42.41cm2.
9 cm
A
B
connection
Centre Angle (ACentre Angle (AooS)S)
xo
National 5
Thursday 20 April 2023Thursday 20 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6565
o o2
51.3x 360
7
o ox 120
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Area Sectorπr2
Sector angle
360o=
Find the angle xo given that the sector area is 51.3mm2.
7 mm
A
B
connection
Centre Angle (ACentre Angle (AooS)S)
xo
20 Apr 202320 Apr 2023 Compiled by Mr. Lafferty Maths Dept.Compiled by Mr. Lafferty Maths Dept.
Now try N5 TJEx13.5
Ch13 (page 130)
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Nat 5
Centre Angle (ACentre Angle (AooS)S)
3 marks