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Transcript of National 5 Thursday, 11 December 2014Thursday, 11 December 2014Thursday, 11 December 2014Thursday,...
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 11
Isosceles Triangles in Circles Right angle in a Semi-Circle
The CircleThe Circlew
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Tangent Line to a CircleDiameter Symmetry in a CircleCircumference of a CircleLength of an ARC of a CircleArea of a CircleArea of a SECTOR of a Circle
Summary of Circle Chapter
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 22
Starter QuestionsStarter Questions
Q3.
Q1. True or false
2Does 8 12 f actorise to (x - 6)(x - 2)x x
Q2. How many degrees in one eighth of a circle.
Q4. 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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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 33
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We are learning to identify isosceles triangles
within a circle.
Aim of Today’s Lesson
Isosceles Triangles Isosceles Triangles in Circlesin Circles
National 5
Tuesday 11 April 2023Tuesday 11 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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in Circlesin Circles
xo xoDEMO
National 5
Tuesday 11 April 2023Tuesday 11 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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in Circlesin Circles
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 66
Q.Q.Find the angle xFind the angle xoo..
A
B
C
Solution
Angle at C is equal to:
Since the triangle is isosceleswe have
360 280 80o o o
xo
280o
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in Circlesin Circles
2 80 180o o ox
2 100o ox
50o ox
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 77
Maths in Action Ex 2.1 page 181
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in Circlesin Circles
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 88
Starter QuestionsStarter Questions
Q3.
Q1. Explain how we solve
2Factorise 13 42 x x
Q2. How many degrees in one tenth of a circle.
Q4. After a discount of 40% a Digital Radio is £120. Explain why the originally price was £200.
5( 2) 20x
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 99
Semi-circle angleSemi-circle angle
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.
Aim of Today’s Lesson
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1010
Semi-circle angleSemi-circle angle
Tool-kit requiredTool-kit required
1.1. ProtractorProtractor
2.2. PencilPencil
3.3. RulerRuler
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1111
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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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1212
Mark three points Mark three points
1.1. Outside the circleOutside the circle
x xx
x x
x x
x
x
Semi-circle angleSemi-circle angle
2. On the 2. On the circumferencecircumference3. Inside the circle3. Inside the circle
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1313
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
Semi-circle angleSemi-circle angle
Log your results in a table. Log your results in a table. InsideCircumferenc
eOutside
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1414
InsideCircumferenceOutside
x
Semi-circle angleSemi-circle angle
x
x
< 90o
> 90o
= 90= 90oo
Begin Maths in Action Book page 182
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DEMO
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1515
Starter QuestionsStarter Questionsw
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Write down as many equations as you can
e.g. a + b = 11
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1616
We are learning to understand what a tangent line is and its
special property with the radius at the point of contact.
Aim of Today’s Lesson
Tangent lineTangent linew
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1717
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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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1818
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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DEMODEMO
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1919
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
SolutionRight-angled at A sinceAC is the radius at the pointof contact with the Tangent.
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
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2020
Tangent lineTangent line
Maths in Action Ex 4.1 page 185
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2121
Starter QuestionsStarter Questions
Q2. 2Factorise 4 9x
Q1. Using FOIL multiply out (x2 + 4x - 3)(x + 1)
Q3. I want to make 15% profit on a computer I bought for £980. How much must I sell it for.w
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Using a multiplication table expand out (x2 + 4x - 3)(x + 1)
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2222
We are learning to understand some special properties
when a diameter bisects a chord.
Aim of Today’s Lesson
Diameter symmetryDiameter symmetryw
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2323
Diameter symmetryDiameter symmetry
A B
C
D
1. A line drawn through the centre of a circle through the midpoint a chord will ALWAYS cut
the chord at right-angles
2. A line drawn through the centre of a circle at right-angles to a chord willALWAYS bisect that chord.
3. A line bisecting a chord at right angles will ALWAYS pass through the centre of a circle.
O
DEMO
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2424
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
Diameter symmetryDiameter symmetry
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
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2525
Maths in Action
Ex 5.1 & Ex 5.2 page 187
Diameter symmetryDiameter symmetryw
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2626
Starter QuestionsStarter Questions
Q1.2 81Factorise x
Q2.12 4 4
True or f alse 28 3 7
Q3. Explain why the area of the triangle is 48m2
Q4.2Solve 2 3x x
12m
8m
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2727
Circumference Circumference of a circleof a circle
Aim of Today’s Lesson
We are learning to use the formula for calculating
the circumference of a circle
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2828
Main parts of the circle
radius
O
CircumferenceC D
Diameter2D r
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of a circleof a circle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2929
Q. Find the circumference of the circle ?
SolutionSolution
C D 8C
25.12C cm
4cm
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of a circleof a circle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3030
Q. The circumference of the circle is 60cm ? Find the length of the diameter and radius.
SolutionSolution
C D 60 D
60D cm
19 D cm
2D r
19 2r192
r
9.5r cm
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of a circleof a circle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3131
Now it’s your turn !
Maths in Action Ex 7.1 page 191
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mCircumference Circumference
of a circleof a circle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3232
Starter QuestionsStarter Questions
Q3.
Q1. True or false
72Simplif y
360
Q2. Using the balancing method rearrange into D =
Q4. Calculate
5(2 1) 2(5 1) x x
C D
2 2117 36
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3333
length of the length of the arc of a circlearc of a circle
Aim of Today’s Lesson
We are learning to use the formula for calculating the length of an arc.
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3434
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
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3535
Q. Find the circumference of the circle ?
SolutionSolution
C D 20C
62.8C cm
10cm
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circlecircle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3636
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
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3737
60
( 18)360
o
oarc length
length 9.42arc cm
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circlecircle
Arc lengthπD
Arc angle
360o=
Q. Find the length of the minor arc AB below ?
9 cm
60o
A
B
connection
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3838
260
( 20)360
o
oarc length
length 45.38arc cm
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circlecircle
Arc lengthπD
Arc angle
360o=
Q. Find the length of the major arc PQ below ?
10 m
100o
P
Q
connection
260o
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3939
Now it’s your turn !
Maths in Action Ex 8.1 page 193
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mlength of the length of the arc of a circlearc of a circle
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4040
Starter QuestionsStarter Questions
Q3.
Q1. True or false
2Does 4 4 f actorise (x - 2)(x - 2)x x
Q2. Expand out (x + 3)(x2 + 40 – 9)
Q4. 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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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4141
The Area of a The Area of a circlecircle
Aim of Today’s Lesson
We are learning to use the formula for calculating
the area of a circle
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4242
23
4
12
3
456
7
8
1 56
78
If we break the circleinto equal sectors
And lay them out side by side We get very close
to a rectangle.
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4343
12 3
45
67
8
thinner and thinnersectors
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.
r
r
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4444
r
r
2Area of a circle r
Area of a rectangle l b 2Area of a rectangle r r r
But the area inside this rectangle is also the area of the circle
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4545
Q. Find the area of the circle ?
SolutionSolution
2A r 24A
250.26A cm
4cm
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4646
The Area The Area of a circleof a circle
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q1-2
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4747
Q. The diameter of the circle is 60cm. Find area of the circle?
SolutionSolution
2A r60
302 2D
r cm
230A22827.43A cm
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4848
The Area The Area of a circleof a circle
Now it’s your turn !
Maths in Action
Ex9.1 page 194
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4949
Q. The area of a circle is 12.64 cm2.
Find its radius?SolutionSolution
2A r
2 12.64
4r cm
4 2r cm
212.64 r
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5050
The Area The Area of a circleof a circle
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q5 onwards
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5151
Starter QuestionsStarter Questions
Q3. Calculate
Q1. Find the missing numbers
2 51 25 8
Q2. Using the balancing method rearrange into x =
24 12 9 (2 ?)(? ?) x x x x
1 y x
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5252
Sector area of a circleSector area of a circle
Aim of Today’s Lesson
We are learning to use the formula for calculating the sector of an circle.
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5353
A
B
major sector
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circlecircle
minor sector
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5454
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
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5555
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
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5656
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
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5757
2260Sector Area ( 10 )
360
o
o
2Area Sector 226.89 cm
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circlecircle
Sector Areaπr2
Sector angle360o
=
Q. Find the area of the major sector PQ below ?
10 m
100o
P
Q
connection
260o
National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5858
Sector area of a circleSector area of a circle
Now it’s your turn !
Maths in Action Ex 10.1 page 196
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National 5
Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5959
Starter QuestionsStarter Questions
Q3.
Q1. Using the balancing method rearrange into x =
2Factorise 3 2 x x
Q2. True or false 2(x - 3) + 3x = 5x - 6
12 10
2x y
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6060
Area is2A r
Summary of Circle Summary of Circle TopicTopic
Circumference is
C D
Sector area
sector2
angleArea =
360
o
o
centrer
Arc length is
lengthArc angle
= D 360
o
o
centre
Diameter
2D rRadiu
s12
r D
line that bisects a chord
1. Splits the chord into 2 equal halves.
2. Makes right-angle with the chord.
3. Passes through centre of the circle
Pythagoras TheoremSOHCAHTOA
Semi-circle angle is always 90
o
Tangent touches circle at one pointand make angle 90
o with point of
contact radius
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Tuesday 11 April 2023Tuesday 11 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6161
Summary of Circle TopicSummary of Circle Topic
Maths in Action Book Page 199Maths in Action Book Page 199
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