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TECHNO PEDAGOGIC CONTENT KNOWLEDGE ANALYSIS (PRACTICUM)TRANSCRIPT
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MATHEMATICS
STANDARD
IX
Government of Kerala
DEPARTMENT OF EDUCATION
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NATIONAL ANTHEM
Jana-gana-mana-adhinayaka,jaya heBharata-bhagya-vidhata.Punjab-Sindh-Gujarat-MarathaDravida-Utkala-BangaVindhya-Himachala-Yamuna-GangaUchchala –Jaladhi-taranga.Tava shubha asisa jage,Tava subha asisa mage,Gahe tava jaya gatha,Jana-gana-mangala-dayaka jaya heBharata-bhagya-vidhata.Jaya he, jaya he, jaya he,Jaya jaya jaya , jaya he!
PLEDGE
India is my country. All Indians are my brothers and sisters. I love my country, and I am proud of its rich and varied heritage. I shall always strive to be worthy of it. I shall give respect to my parents, teachers and all elders and treat everyone with courtesy. I pledge my devotion to my country and my people. In their well-being and prosperity alone lies my happiness
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CONTENTS
PerimeterPerimeter and diameterCircles and polygonsA new numberArcArcs and anglesLength of an arcAreaSectors
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Chapter - 11
Circular Measures
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Perimeter
The perimeter is the sum of the length of all sides of a closed figure.
What is the perimeter of a square of side 3cm?
The perimeter is 3+3+3+3=12cm.
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How do we find the perimeter of a circle of diameter 3cm?
We cannot compute it as in the case of a square;
We can place a string around it ,straighten and measure.
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perimeter and diameter
When the diameter is increased,the perimeter also increases.
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Circles and polygons
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A new numberThe perimeter of a circle is proportional
to its diameter.The perimeter of any circle divided by it
diameter must give the same number.Actually this number is irrational . In
fact there is a special symbol in mathematics for this number . This number pi.
perimeter of circle diameter of circle
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Arcs
An arc is a portion of circle. P QA BAB and PQ are parts of a circle.Usually ,we write AB or PQ to
denote the line joining two points.
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Arcs and angles
In the figure below , ABP is an arc of the circle. A PSuppose the point P moves away from A,along
the circle. A p A
A
B
B
B
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Then the length of the arc ABP also increases.
A B P
O O
A
BP
As the length of the arc ABP increases , so does the angle AOP At the center of the circle
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Central angle
The angle made by joining the end points of an arc to the centre of the circle is called the central angle of the arc.
AB
P
O A
B
P
O
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O
A
B
P
The central angle of the arc ABP is now 180 degree .suppose P goes down45 degreeMore , the central angle of ABP is 225 degree. When P moves 45 degree more to the Right the central angle becomes 270 degree . Another 45 degree up and it becomes
315 degree.45 degree more upwards and ABP becomes the full circle.
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Length of an arcIf the radius of a circle is denoted by r,its
perimeter is 2 r.So the length of an arc of central angle 1 is
1\360 of perimeter=2 r*1\360 arc length=2\360 of perimeter=2
r*2\360For an arc of central angle 1\2,Arc length=1\2*1\360 of perimeter=2
r*1\2*1\360.In general,for an arc of central angle x. 2 r*x*1\360=2 r*x\360.
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The length of an arc of a circle is that part of the perimeter of the circle,as the central angle is of 360.
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Area
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Draw regular polygons with more and more sides with in the circle , their areas would get
closer and closer to the area of the circle.
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By joining the vertices of the polygon to the center of the circle , we can divide the polygon into triangle
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If the polygon has n sides, then its area is :n*(1/2)sh= (1/2)nsh‘ns’ is the sum of all sides of the polygon
In other words ns is perimeter
Perimeter denoted by p
Area of the polygon = (1/2)ph
Increase the number of sides , the polygon gets closer and closer to the circle . The perimeter and area of the polygon get closer and closer to the
perimeter and area of the circle
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Area of the circle = (1/2)*perimeter of the circle * radius
= (1/2)*2∏*radius*radius
= ∏*square of radius
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Sectors
In the figure below,two points of a circle are joined to the centre.
The figure obtained thus is called a sector of the circle.
Thus a sector is formed by an arc of a circle and the radii through its end points.
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LOOK AT THE PICTURE
As the central angle increases, so does the area of the sector. We can show that the area of a sector of central angle x is x/360 of the area of the whole circle.
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The area of a sector of a circle is that part of the area of the circle as the central angle is of 360 degree.
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Area of a sector
In a circle of radius r, a
sector of central angle x has Area,
πr2 x x
360
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THANK YOU