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How perimeter of segment of a circle is πrθ180+2rsin(θ2). Give derivation.

Answer
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Hint: We will first of all create a circle of radius r and a sector. Then join the end points of the sector by a line to form a chord and thus a segment. Now, to find the perimeter of the segment you just need to add the length of the chord and the arc length.

Complete step-by-step answer:
Let us first draw a circle with radius r and a sector with angle AOB=θ and then by joining the end points A and B of the sector, we will get a chord AB as given below:
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Here, O is the center, AB is the chord and C is a point on the arc ACB.
Now, without loss of generality, let us say we have the segment ACB to find the perimeter of.
To find its perimeter, we definitely need to add the length of the arc and the length of the chord as well.
We know that the length of an arc with angle θ is given by θ360×2πr=θπr180……….(1)
Now, let us try to find the length of the chord AB. Draw a perpendicular from point O to AB.
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Since, we know that we have a theorem: Perpendicular from center to chord bisects the chord.
Hence, AD = DB. ………………(2)
Now, consider AOD and BOD:
AD = DB (Using 2)
OD = OD (Common)
OA = OB (Radii of same circle)
Hence, AODBOD (By SSS property rule)
[SSS Rule: If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent]
Hence, AOD=BOD (By CPCT)
[CPCT stands for Corresponding parts of congruent triangles. CPCT theorem states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other]
Hence, if AOB=θ, then AOD=BOD=θ2.
Now, in AOD, we have sin(θ2)=ADOA
On cross multiplying and simplifying and putting in OA = r, we get:-
AD=rsin(θ2)
Now, using (2), we get: AD=DB=rsin(θ2)
Adding both, we get:-
AD+DB=2rsin(θ2)
We can rewrite it as:-
AB=2rsin(θ2) ……………(3)
Adding both (1) and (3) to get the perimeter:-
Perimeter of segment ADBC = θπr180+2rsin(θ2).
Hence, proved.

Note: The students must note that the arc length formula, we just wrote can be visualized as well instead of learning only. Since, everyone knows that the circumference of a circle is given by 2πr .
We also know that the angle in a circle is 360.
It means for 360, the boundary is 2πr.
So, for 1, the boundary is given by 2πr360.
Hence, for θ, the boundary will be 2πr360×θ=πrθ180.