The center of the circle is 0, and RS = ST = 4. What is the length of

According to me,

Join OS ,OR and OT
we know OS =OR=OT=r

Consider triangle ORS
OR=OS =r
=> Angle ORS= Angle OSR ( angles opp to equal sides)

Similarly in triangle OST
OS=OT=r
=> Angle OST= OTS

But OS=OR=OT
=> angle ors= angle osr = angle ost = angle ots

Given=> Angle RST = 120
=> as angle OST = angle OSR
Angle OST = 60

=> in triangle OST
ANGLE OST = ANGLE OTS ( as proved above)
Angle ots =60
=> angle sot =60 ( by angle sum prop of triangle)
Therefore triangle OST is an equilateral triangle as all angles are equal
=> os=ot=st =4

Similarly Triangle ORS Is equilateral triangle
=>rs=or=os =4

Now , calculating

For arc RWT
(Angle /360)* 2*pi*r=( 240/360) *2* pi* 4
= > 16Pi/3
Therefore option C

Correct me if I am wrong, otherwise Kudos.

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radius = 4 = RS= ST
angle = 120 *2 ; 240
2 * 4* pi * 240/360 = \(\frac{16 \pi}{3}\)
IMO C

I have forgotten the formulas of circles. Can anyone explain the answer to me?

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How do you know that R is the radius in this case?
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Can anybody explain this one , how is the radius = 4 ?
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How have you considered radius to be 4?

Posted from my mobile device

From the image below Quadrilateral RWTS is cyclic quadrilateral (CQ) and in a CQ sum of opposite angles is 180
∴ ∠RWT + ∠RST = 180
∠RWT = 180-120 = 60

Another property, if angle subtended by arc RST at the center is θ then the angle subtended in the remaining arc will be θ/2. The opposite is also true.
Since, angle made by arc RST in the remaining arc is 60 degree then the angle subtended at the center by the same arc is 2*60 = 120

And △s ROT and RST are isosceles triangles and if we drop a perpendicular from the point of intersection of two equal sides then the perpendicular:
1. Bisects the angle
2. Bisects the base
3. Perpendicular to the base

Therefore, we can see that △s ROS and OST are equilateral △s as all the angles are 60 degree and since, RS = 4 therefore radius = OR = OS = 4

Length of arc RWT = 2πrθ/360

θ = 360-120 = 240
r = 4

Length of arc RWT = 2π*4*240/360 = 16π/3 (C)