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length of line segment RU

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by alex.gellatly » Tue Jun 05, 2012 5:44 am
The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is {4pi}/3, what is the length f line segment RU?

A. 4/3
B. 8/3
C. 3
D. 4
E. 6

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by Jim@StratusPrep » Tue Jun 05, 2012 9:22 am
The length of the circumference is Pi times diameter or 8pi in this case

8pi/6 = 4pi/3, so the arc is 1/6th of the circumference, which means the central angle is 1/6 or 360 or 60 degrees. Since the other 2 sides must be equal, this is an equilateral triangle. As the other sides are radii, their length is 4, which is the same as RU

Answer D.
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by Anurag@Gurome » Tue Jun 05, 2012 7:47 pm
alex.gellatly wrote:The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is {4pi}/3, what is the length f line segment RU?

A. 4/3
B. 8/3
C. 3
D. 4
E. 6

Thanks
Circumference of the circle = 2(pi)(4) = 8(pi)
Let x be the angle subtended at the center of circle.
Then (x)(8)(pi)/360 = 4(pi)/3
x = 60, now RC and RU are equal so angles CRU and RUC should be equal. Hence, triangle RCU is an equilateral triangle with each angle 60º.

So, RU = 4

The correct answer is D.
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by alex.gellatly » Sat Jul 07, 2012 10:45 pm
Jim@StratusPrep wrote:The length of the circumference is Pi times diameter or 8pi in this case

8pi/6 = 4pi/3, so the arc is 1/6th of the circumference, which means the central angle is 1/6 or 360 or 60 degrees. Since the other 2 sides must be equal, this is an equilateral triangle. As the other sides are radii, their length is 4, which is the same as RU

Answer D.
Why must the other 2 sides be equal?

I'm sorry... but I still don't understand this question. Could someone explain differently?
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by eagleeye » Sat Jul 07, 2012 11:01 pm
alex.gellatly wrote:
Jim@StratusPrep wrote:The length of the circumference is Pi times diameter or 8pi in this case

8pi/6 = 4pi/3, so the arc is 1/6th of the circumference, which means the central angle is 1/6 or 360 or 60 degrees. Since the other 2 sides must be equal, this is an equilateral triangle. As the other sides are radii, their length is 4, which is the same as RU

Answer D.
Why must the other 2 sides be equal?

I'm sorry... but I still don't understand this question. Could someone explain differently?
Thanks
I am guessing you are ok till the part of upto the central angle being 60 degrees.

Now the other two sides are both radii of the circle. Hence they are both equal to 4. So we have one angle = 60, the other two sides (which are two radii) = 4.

If two sides are equal and one angle is 60, the other two angles must be 60 as well. (2x+60 = 180 => 2x=120 => x=60). Since all three angles are 60, it is an equilateral triangle. So all three sides are equal. Since the other 2 sides are the radii each of length 4, the third one is 4 also.

Did this help ?
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by alex.gellatly » Sat Jul 07, 2012 11:52 pm
eagleeye wrote:
alex.gellatly wrote:
Jim@StratusPrep wrote:The length of the circumference is Pi times diameter or 8pi in this case

8pi/6 = 4pi/3, so the arc is 1/6th of the circumference, which means the central angle is 1/6 or 360 or 60 degrees. Since the other 2 sides must be equal, this is an equilateral triangle. As the other sides are radii, their length is 4, which is the same as RU

Answer D.
Why must the other 2 sides be equal?

I'm sorry... but I still don't understand this question. Could someone explain differently?
Thanks
I am guessing you are ok till the part of upto the central angle being 60 degrees.

Now the other two sides are both radii of the circle. Hence they are both equal to 4. So we have one angle = 60, the other two sides (which are two radii) = 4.

If two sides are equal and one angle is 60, the other two angles must be 60 as well. (2x+60 = 180 => 2x=120 => x=60). Since all three angles are 60, it is an equilateral triangle. So all three sides are equal. Since the other 2 sides are the radii each of length 4, the third one is 4 also.

Did this help ?
Sorry.. but I still don't get it. I understand how and why the central angle must be 60. However, I don't understand why the lengths are equal to R. Maybe if I had a visual representation or something. I've tried to draw it out... but I'm not following.
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by eagleeye » Sun Jul 08, 2012 12:13 am
alex.gellatly wrote:
eagleeye wrote:
alex.gellatly wrote:
Jim@StratusPrep wrote:The length of the circumference is Pi times diameter or 8pi in this case

8pi/6 = 4pi/3, so the arc is 1/6th of the circumference, which means the central angle is 1/6 or 360 or 60 degrees. Since the other 2 sides must be equal, this is an equilateral triangle. As the other sides are radii, their length is 4, which is the same as RU

Answer D.
Why must the other 2 sides be equal?

I'm sorry... but I still don't understand this question. Could someone explain differently?
Thanks
I am guessing you are ok till the part of upto the central angle being 60 degrees.

Now the other two sides are both radii of the circle. Hence they are both equal to 4. So we have one angle = 60, the other two sides (which are two radii) = 4.

If two sides are equal and one angle is 60, the other two angles must be 60 as well. (2x+60 = 180 => 2x=120 => x=60). Since all three angles are 60, it is an equilateral triangle. So all three sides are equal. Since the other 2 sides are the radii each of length 4, the third one is 4 also.

Did this help ?
Sorry.. but I still don't get it. I understand how and why the central angle must be 60. However, I don't understand why the lengths are equal to R. Maybe if I had a visual representation or something. I've tried to draw it out... but I'm not following.
Image

Please see attached. [/img]
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