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22 Dec 2005, 00:54
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
63% (01:33) correct
37%
(01:33)
wrong
based on 372
sessions
History
Date
Time
Result
Not Attempted Yet
What is the circumference of the circle above?
(1) The length of arc XYZ is 18.
(2) r = s
Attachment:
2022-10-02_20-12-28.png [ 8.06 KiB | Viewed 5484 times ]
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
22 Dec 2005, 01:29
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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.
- 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.
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PastedImage.jpg [ 8.57 KiB | Viewed 14343 times ]
But what if it is like this?
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possibility.JPG [ 6.19 KiB | Viewed 14233 times ]
30 Jan 2011, 12:08
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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
\(Circumference=2*\pi*radius\)
\(\pi\) 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
