the circle
DESCRIPTION
The Circle. Isosceles Triangles in Circles. Right angle in a Semi-Circle. Tangent Line to a Circle. Diameter Symmetry in a Circle. Circumference of a Circle. www.mathsrevision.com. Length of an ARC of a Circle. Area of a Circle. Area of a SECTOR of a Circle. Summary of Circle Chapter. - PowerPoint PPT PresentationTRANSCRIPT
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Isosceles Triangles in Circles Right angle in a Semi-Circle
The CircleThe Circlew
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srevis
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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
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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
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Special Properties of Isosceles TrianglesSpecial Properties of Isosceles Triangles
Two equal lengthsTwo equal lengths
Two equal anglesTwo equal angles
Angles in any triangle sum to 180Angles in any triangle sum to 180oo
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
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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
2 80 180
180 80 2
2 100
50
o o o
o o o
o o
o o
x
x
x
x
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mIsosceles triangles Isosceles triangles
in Circlesin Circles
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Tangent lineTangent line
A A tangent linetangent 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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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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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 have2 2 2
2 2 2
2 2 2
2
8 10
10 8
100 64 36
36 6
a b c
a
a
a
a
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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
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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 2
2 2 2
2 2 2
2
6 10
10 6
100 36 64
64 8
a b c
a
a
a
a
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
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Main parts of the circle
radius
O
CircumferenceC D
Diameter2D r
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mCircumference Circumference
of a circleof a circle
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Q. Find the circumference of the circle ?
SolutionSolution
C D 4C
12.56C cm
4cm
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of a circleof a circle
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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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mCircumference Circumference
of a circleof a circle
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Q. What is an arc ?
A
B
AnswerAnswer
An arc is a fractionof the circumference.
minor arc
major arcww
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mlength of the length of the arc of a circlearc of a circle
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Q. Find the circumference of the circle ?
SolutionSolution
C D 20C
62.8C cm
10cm
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mlength of the length of the arc of a circlearc of a circle
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Q. Find the arc length for the semi-circle ?
Solution
C D length of circle circumf erencearc Fraction
31.4arc length cm
1
length ( )2
arc D180o
1
length ( 20)2
arc
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Q. Given part of the circle below. Find the arc length ?Solution
C D
1
( )4
arc length D
length 15.7arc cm
10cm length of circle circumf erencearc Fraction 90
o
1
( 20)4
arc length
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mlength of the length of the arc of a circlearc of a circle
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3 step process to calculating the length of an arc.
Step 1 Find the circumference of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the arc length.
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mlength of the length of the arc of a circlearc of a circle
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Solution
C D
1
( 12)8
arc length
length 4.71arc cm
6cm
length of circle circumf erencearc Fraction
45o
Q. Part of a circle is given below. Find the arc length.
45 1f raction=
360 8
o
o
Fraction of
circle
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mlength of the length of the arc of a circlearc of a circle
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3 step process to calculating the length of an arc.
Step 1 Find the circumference of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the arc length.
C D
at centre360o
angle
length of circle circumf erencearc Fraction
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Q. Find the length of the arc AB below ?
9 cm
60o
A
B
SolutionSolution
1 : C = D
= 18
= 56.52cm
Step
60 1 2 : Fraction=
360 6Step
1 3 : arc length= 56.52 9.42
6Step cm
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mlength of the length of the arc of a circlearc of a circle
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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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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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Q. What is an sector area ?
Sector area of a circleSector area of a circle
A
B
AnswerAnswerSector area is a fraction
of the area of the whole circle.
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major sectorww
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Sector area of a circleSector area of a circle
Q. Find the area of the circle ?
SolutionSolution
2A r 210A
2314A cm
10cm
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Q. Find the sector area for the semi-circle ?
Solution
2A r of circle Area of circleSector Fraction
2157Sector cm
Sector area of a circleSector area of a circle
21( )
2Sector r180
o
21( 10 )
2Sector
10cm
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Q. Given part of the circle below. Find the sector area ?Solution
2A r
21( )
4Sector r
27.85Sector cm
Sector area of a circleSector area of a circle
c of circle Area of circleSe tor Fraction
21c ( 10 )
4Se tor
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Sector area of a circleSector area of a circle
3 step process to calculating sector area of a circle.
Step 1 Find the area of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the sector area.
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Solution
2A r
1
( 36)8
Sector
2Sector 14.13cm
6cm
Sector area of a circleSector area of a circle
of circle area of circleSector Fraction
45o
Q. Part of a circle is given below. Find the sector area.
45 1f raction=
360 8
o
o
Fraction of
circle
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Sector area of a circleSector area of a circle
3 step process for calculating the sector area.
Step 1 Find the area of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the sector area.
2A r
at centre360o
angle
ec of circle area of circleS tor Fraction
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Q. Find the area of the sector below ?
Sector area of a circleSector area of a circle
SolutionSolution
2
2
1 : A=
= 81
=254.34cm
Step r
60 1 2 : Fraction=
360 6Step
21 3 : arc length= 254.34 42.39
6Step cm
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Area is2A r
Summary of Circle Summary of Circle TopicTopic
Circumference is
C D
Sector area2 angle
Sector = 360
o
o
centrer
Arc length is
length
angleArc =
360
o
o
centreD
Diameter2D r
Radius12
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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