The length of arc XYZ is the sum of the below two arcs
- the length of arc XY
- the length of arc YZ

Statement1)

We need to know degree of s in order to find out the length of the arc XY or the arc YZ.

Since we do not know s, statement 1 alone is not sufficient.

Statement2)

If r = s, then r = s = 60. However, we do not know the radius of the circle, and so we cannot find out the circumference of the circle.

Thus statement2 alone is not sufficient.

Statement1 and 2)

Since r = s = 60 and the length of the arc XY is 9, we know the following;

- The length of the arc XY is 9.
- The length of the arc YZ is 9.
- The length of the arc XZ is 9.
--------------------------------
The circumference of the circle is 27.

Thus statement 1 and 2 combined are sufficient.

But what if it is like this?

good question...

Ans. C

Stat 1 is insufficient, does not provide enough info to determine the circumference.

Stat 2 suggest, the triangle is equilayeral so all the side are same in lenght, therefore all the 3 arc of the circle is same.... insufficient,

However combining togethe, it is possible to determine the circumferene, in this case it is 27.

Can you please label the angle and points?

Uhm, i don't know whether there will normally be "not drawn to scale" to indicate that the figure are not exactly what we perceive ...Anyways, you're right to stem such a possibility!! . In that case, your original answer of E is correct.

Thanks for the labels

in this case also the answer is C.

Stat 1 is insufficient, to calculate the circumference

statement 2 suggest the traingle is equilateral and hence r=60o
therefore the arc xz = 1/3 of the circumference.

combining together stat 1&2, circumference can be determined

well.. the measure of arc of a circle is double the angle it (arc) forms.

in this case 60o , arc=120, and hence 120/360 =1/3 of circumference

Sorry, I was double-minded but this time, i'm sure it's C and i'm gonna prove why

what matters here is that the angle of the arc XYZ is constant!

Can you please explain the meaning of arc XYZ to me?

Is it the edge from X via Y to Z?

Or the arc opposite of Y?

Thx

AGREE!!!

Thank you folks!

Combining (1) and (2), it follows that arc XY=ZY=ZX (they are fomed by equal angles)

===> XY=(XY+ZX)/2=9

Circumference=sum of 3 arcs=3*9=27

uhm, this we're not sure as vivek pointed out in his illustration

Thanks, I reviewed you attachment. Very nice. But I believe The following principle.....

We have : XOY= 2 *XZY coz this is a principle in geometry.

....applies only if the point O is a point of intersection of three perpendicular bisectors. We don't know if it is. Can we really apply this principle in that case?

Yes, O is coz O is the centre of the circle! ...the distances from O to X, Y and Z are the same(= the radius) --> O lies on the perpendicular bisectors of XY, YZ and XZ.

Q: What is the radius of circle.
\(Circumference=2*\pi*radius\)
\(\pi\) is a constant.

1. \(\stackrel{\frown}{XYZ}=18\)
\(\stackrel{\frown}{XYZ}=(\theta/360)*2*\pi*radius\)
\(\stackrel{\frown}{XYZ}\)=length of the arc XYZ
\(18=(\theta/360)*2*\pi*radius\)
\(\theta\) angle subtended by major arc \(\stackrel{\frown}{XYZ}\) at the center of the circle
\(\theta\) unknown
NOT SUFFICIENT.

2. \(\angle r = \angle s\)
The triangle is an equilateral triangle and \(\angle r = 60^{\circ}\)
Minor arc \(\stackrel{\frown}{XZ}\) makes an angle of \(60^{\circ}\) with the point Y, which lies on the circumference.
Theorem,The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
So if the minor arc XZ is making an angle of r with a point on the circumference of the circle, minor arc XZ will make an angle of 2r with the centre of the circle. Now we know angle of minor arc XZ is \(2r=120^{\circ}\)
And angle subtended by major arc XYZ at the center is \(\theta=360-120=240\)
However, with this information also we won't be able to find the radius of the circle because the length of the arc is not known. NOT SUFFICIENT.

Combining both of the above;
We know, length of major arc \(\stackrel{\frown}{XYZ}\) and angle made by the arc from the center i.e. \(\theta\)

Put these into the formula now;
\(\stackrel{\frown}{XYZ}=(\theta/360)*2*\pi*radius\)
\(\theta=240^{\circ} \hspace \hspace \stackrel{\frown}{XYZ}=18\)
\(18=(240/360)*2*\pi*radius\)
\(radius=27/(2*\pi)\)

\(Circumference=2*\pi*radius=(2*\pi)*27/(2*\pi)=27\)
SUFFICIENT.


Answer: C
OA: C

What is the circumference of the circle above?

(1) The length of arc XYZ is 18.
(2) r = s

In order to solve this question, we need to know the radius.

Statement (1) allows us to parse out a formula we need to solve the problem- x/360 times the circumference of the circle or 2pir equals 18; however, there are two variables so we cannot solve this question.

Statement (2) implies an equilateral triangle- we can further the inscribed angle theorem; however, we need a radius length in order to calculate the circumference- we still have two variables

Statement (1) and (2) together are sufficient because they provide each other's missing variable. Hence, Statement (1) and (2) are sufficient.

I understand the first one to be sufficient because an arc is 2/3 of circumference of a circle.
Can someone explain why the second statement is sufficient.

The correct answer is C, not D. Please check solution here: https://gmatclub.com/forum/what-is-the- ... ml#p860848
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