Can someone help me on the below geometry problem with detailed steps?
What is the circumference of the circle below (image attached)?
(1) The length of arc XYZ is 18.
(2) r = s
Geometry problem  Curcumference of a circle
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1. not sufficient > we cannot use given arc lenght because it might vary for different angles
2. not sufficient  > not sufficient data given to compute cir.
Both 1 & 2 sufficient  > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,
2/3 * Circumference = 18 > Cir = 27.
IMO, the answer is C.
What is the OA ?
2. not sufficient  > not sufficient data given to compute cir.
Both 1 & 2 sufficient  > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,
2/3 * Circumference = 18 > Cir = 27.
IMO, the answer is C.
What is the OA ?

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I am really flummoxed. Please help! How can r=s along with (i) help to solve the question. Isn't r radius and s the angle YXZ and YZX?sasi78 wrote:1. not sufficient > we cannot use given arc lenght because it might vary for different angles
2. not sufficient  > not sufficient data given to compute cir.
Both 1 & 2 sufficient  > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,
2/3 * Circumference = 18 > Cir = 27.
IMO, the answer is C.
What is the OA ?
I am missing something...
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I'm in the same boat I don't understand why it would mention anything about r. I thought the y was r, if it was then you would have enough info to solve the equation.kanha81 wrote:I am really flummoxed. Please help! How can r=s along with (i) help to solve the question. Isn't r radius and s the angle YXZ and YZX?sasi78 wrote:1. not sufficient > we cannot use given arc lenght because it might vary for different angles
2. not sufficient  > not sufficient data given to compute cir.
Both 1 & 2 sufficient  > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,
2/3 * Circumference = 18 > Cir = 27.
IMO, the answer is C.
What is the OA ?
I am missing something...
What does r stand for? Radius? Why would it give an angle measure equal to a length?
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The drawing misled you guys, since it's really small. That "y" you see on the picture is actually an "r" and it's just a random letter used to designate angle XYZ.
The notation is: r is the measurement of angle XYZ.
As you may imagine, you can't have r as radius, since it's impossible to say "length" = "measurement of angle" (it would be like comparing horses to chicken) in statement 2. r has to be an angle as well, since you only compare angles to angles.
The notation is: r is the measurement of angle XYZ.
As you may imagine, you can't have r as radius, since it's impossible to say "length" = "measurement of angle" (it would be like comparing horses to chicken) in statement 2. r has to be an angle as well, since you only compare angles to angles.

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This might be a silly question, but "how" or "why" do you multiply the circumference by 2/3?sasi78 wrote:1. not sufficient > we cannot use given arc lenght because it might vary for different angles
2. not sufficient  > not sufficient data given to compute cir.
Both 1 & 2 sufficient  > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,
2/3 * Circumference = 18 > Cir = 27.
IMO, the answer is C.
What is the OA ?
Isn't 
arc length = 18 = (n/360) * 2 * pi * r, where n=60
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hey kanha18,
if you see the circle, it is made of three arcsXY, YZ, XZ
statement 2 says r=s given that the base angles (s) are equal, that makes the triangle XYZ an equilateral triangle (easy to see if you join XZ, angle YXZ=s and angle YZX=s, r=s)
so all arcs would be equal in length=>XY=YZ=XZ
(XY+YZ+XZ=circumference)
so XY+XZ forms 2/3 of the circumference
if you see the circle, it is made of three arcsXY, YZ, XZ
statement 2 says r=s given that the base angles (s) are equal, that makes the triangle XYZ an equilateral triangle (easy to see if you join XZ, angle YXZ=s and angle YZX=s, r=s)
so all arcs would be equal in length=>XY=YZ=XZ
(XY+YZ+XZ=circumference)
so XY+XZ forms 2/3 of the circumference