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Find the area of the minor segment of a circle of radius 42 cm, if the length of the corresponding arc is 44 cm.

Answer
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Hint: In this sum, we will first draw figure for better understanding. After that, we will find angles subtended by an arc of the given length. For finding the area of the segment, we need to find areas of sector and area of a triangle formed. We can find the area of the sector by using angles found. For finding the area of the triangle we will use basic formula but for finding base and height we will use congruency, right-angled triangle properties.

Complete step-by-step solution
Formulas which we will use are:
I) Length of arc=θ360×2πr where r is radius of circle.
II) Area of segment = Area of the sector - Area of a triangle.
III) Area of sector=θ360×πr2 where r is radius of circle.
IV) Area of triangle=12×base×height
Let us draw the figure first:
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Now, let us find angle subtended by arc with help of length of arc 44 cm, radius 42 cm. As we know,
Length of arc=θ360×2πr
Putting the values we get:
44=θ360×2π×4244=θ360×2×227×4244×360×72×22×42=θθ=60(1)
As we can see from diagram, area of segment APB = Area of sector OAPB - Area of triangle OAB . . . . . . . . . (2) so, let us first find area of sector OAPB and area of triangle OAB. Given radius r = 42 cm.
Area of sector OAPB=θ360×πr2θ360×227×(42)2
Putting value of θ from (1), we get:
60360×227×42×42924cm2
Area of sector OAPB=924cm2
Area of triangle AOB=12×base×height
But we don't know values of base or height. So let us find that,
Draw OMAB
Therefore, OMB=OMA=90
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In ΔOMA and ΔOMB we can see that OMB=OMA=90
OA = OB as both is the radius of the same circle.
OM = OM which is common in both triangles.
Therefore, by RHS congruency criterion,
ΔOMA=~ΔOMB
By using CPCT (corresponding parts of congruent triangle) we can say,
AOM=BOM and BM=AM
Therefore, AOM=BOM=12BOA
As we have find θ=BOA=60
Therefore, AOM=BOM=12×60=30
Also, BM=AM=12AB(3)
Now from ΔOMA
sin30=side opposite of angle 30hypotenuse12=AMAO12=AM42AM=21
Also,
cos30=side adjacent of angle 30hypotenuse32=OMAO32=OM42OM=213
From (3) AB = 2AM
Therefore, putting value of AM, we get:
AB = 42 cm
Also, OM=213cm
So, we have found length of base and perpendicular for triangle AOB.
Area of triangle AOB=12×AB×OMArea of triangle AOB=12×42×213Area of triangle AOB=4413cm2
From (2) as we know, area of segment APB = area of sector OAPB - area of triangle OAB.
We have, Area of segment=(9244413)cm2

Note: Students should not get confused with length of arc or length of chord such that, by two points. Students should always draw diagrams for better understanding. Also, don't get confused with formulas of sinθ and cosθ. Don't forget to put units after finding a value for area use squared units and length as linear units.

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