A circle with radius r cm and the angle subtended at the centre of the circle is x°. What is the formula for finding the length, acm, of this arc?
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Answer:
⇒ Perimeter of the sector =50cm
⇒ r is radius of a sector and θ is angle of the sector.
⇒ Perimeter of the sector =
360
o
θ
×2πr+2r
⇒ 50=
360
o
θ
×2πr+2r
⇒ 50−2r=
360
o
θ
×2πr
∴ 2(25−r)=
360
o
θ
×2×πr
∴
πr
(25−r)×360
o
=θ
∴ θ=
π
360
o
(
r
25
−
r
r
)
∴ θ=
π
360
o
(
r
25
−1)
⇒ Area of a sector A=
360
o
θ
×πr
2
Now, substituting value of θ we get,
⇒A=
360
o
π
360
o
(
r
25
−1)
×πr
2
⇒ A=
π×360
o
360
o
(
r
25
−1)×πr
2
⇒ A=r
2
(
r
25
−1)
∴ A=25r−r
2
Hence Proved
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