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The points R, T, and U lie on a circle that has radius 4 [#permalink]
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05 Nov 2009, 08:03
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The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is \(\frac{4*\pi}{3}\), what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
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Last edited by Bunuel on 14 Nov 2013, 13:10, edited 2 times in total.
Edited the question and added the OA



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Re: circle arc [#permalink]
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Last edited by Bunuel on 05 Nov 2009, 12:03, edited 3 times in total.



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Re: circle arc [#permalink]
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05 Nov 2009, 11:33
The diameter is 8 The length of arc formed by the diameter is 4*Pi
the length of arc RTU is 4*PI/3
let the length of the cord is x
So we have a equation
8/4*Pi=x/(4*Pi/3)
So the length of the cord is 8/3
So IMO answer is B



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Re: circle arc [#permalink]
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05 Nov 2009, 11:54
Bunuel  improved some math in your reply, but check that I did not mess it up accidentally.
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Re: circle arc [#permalink]
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05 Nov 2009, 12:34
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jade3 wrote: The diameter is 8 The length of arc formed by the diameter is 4*Pi
the length of arc RTU is 4*PI/3
let the length of the cord is x
So we have a equation
8/4*Pi=x/(4*Pi/3)
So the length of the cord is 8/3
So IMO answer is B The ratio (red part) is not correct. The ratio of the length of chord and arc of it is not the constant value for the given radius circle. The chord length\(=2r*sin(\frac{c}{2})\) where c is the central angle. As the angle by the chord \(\frac{4*\pi}{3}\) is 60 degrees, the length of chord would be: \(=2r*sin(\frac{60}{2})=2r*\frac{1}{2}=r=4\) bb wrote: Bunuel  improved some math in your reply, but check that I did not mess it up accidentally. Thanks bb, you're right it's much easier to read now.
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Re: circle arc [#permalink]
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13 Oct 2010, 10:53
Formula of Length of Arc=(2*Pi*r)*(Angle by arc/360) Length of the arc given= (4*pi)/3
Equate the formula of length of arc
=> (2*Pi*r)*(Angle by arc/360)=(4*pi)/3 => Angle by arc= 60 degrees
=>Since OU=OR(radius of the circle) and angle of the arc at center is 60 degrees.
so angle CUR= angle CRU..hence it is an equilateral triangle and the chord length will be equal to the other two sides(Radius)
Answer is D.



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Re: PS Geometry (Circles/Arcs) [#permalink]
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27 Oct 2010, 10:32
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If you see, the perimeter is 8pi. Length of arc is 4pi/3 = perimeter of the circle/6
So the angle created at center is 360/6 = 60 degree Hence the triangle RTU is an eqilateral triangle.
Hence RU = RT = 4 . Option D
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Re: PS Geometry (Circles/Arcs) [#permalink]
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27 Oct 2010, 10:33
Let's call the center of the circle O; we then have a triangle ORU. Notice that OR and OU are both a radius, so they are equal in length, so this triangle must be isosceles, and the angles at R and U must be equal. The circumference of the circle is 8*Pi. So if arc RTU is 4*Pi/3, then arc RTU is 1/6th of the circle. Thus the angle ORU is 1/6th of 360 degrees, so is 60 degrees. Now the angles at R and U must be equal, and since the angles in this triangle must add to 180, the angles at R and U must both be 60 degrees. So ORU is in fact equilateral, and every side is 4 long.
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Re: circle arc [#permalink]
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06 Dec 2010, 22:37
Bunuel wrote: kirankp wrote: The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*PI/3, what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
this one might be easy.. but i am not able to figure it out.. The circumference of a circle=\(2*\pi*r=8*\pi\), \(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). > Angle \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle). RCU is isosceles triangle as \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\). Answer: D. Bunuel if I follow you correctly RCU is 60 degrees because the arc RTU is 1/6 of the circumference so RCU is the central angle and will have the same measure as the arc. Secondly since RCU=60 degrees as RC and CU are both equal we need to have 180 degrees total in the triangle so we have 120 remaining which is divided by two indicating that all 3 lines have a length of 4? Please advise. thanks.



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Re: circle arc [#permalink]
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gettinit wrote: Bunuel wrote: kirankp wrote: The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*PI/3, what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
this one might be easy.. but i am not able to figure it out.. The circumference of a circle=\(2*\pi*r=8*\pi\), \(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). > Angle \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle). RCU is isosceles triangle as \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\). Answer: D. Bunuel if I follow you correctly RCU is 60 degrees because the arc RTU is 1/6 of the circumference so RCU is the central angle and will have the same measure as the arc. Secondly since RCU=60 degrees as RC and CU are both equal we need to have 180 degrees total in the triangle so we have 120 remaining which is divided by two indicating that all 3 lines have a length of 4? Please advise. thanks. Yes. From RC=CU=r and <RCU=60 we can get that triangle RCU is equilateral.
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Re: circle arc [#permalink]
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07 Dec 2010, 17:55
Excellent, thanks for the confirmation Bunuel!



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Re: circle arc [#permalink]
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02 Oct 2013, 15:16
Bunuel wrote: kirankp wrote: The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*PI/3, what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
this one might be easy.. but i am not able to figure it out.. The circumference of a circle=\(2*\pi*r=8*\pi\), \(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). > Angle \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle). RCU is isosceles triangle as \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\). Answer: D. Can you give me the geometric logic of finding angle RCU (360/6)
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Re: circle arc [#permalink]
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03 Oct 2013, 01:28
honchos wrote: Bunuel wrote: kirankp wrote: The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4*PI/3, what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
this one might be easy.. but i am not able to figure it out.. The circumference of a circle=\(2*\pi*r=8*\pi\), \(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). > Angle \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle). RCU is isosceles triangle as \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\). Answer: D. Can you give me the geometric logic of finding angle RCU (360/6)Arc RTU is 60 degrees, thus the central angle RCU, which subtends it, is also 60 degrees (a central angle in a circle determines an arc).
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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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26 Jul 2014, 03:06
Bunuel, I am an avid follower of you. Would you pls draw the diagram according to the question and how you solved it. I haven't got it yet.



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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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05 Aug 2014, 05:12
If you are running low on time  consider the following:
You know RU is a bit shorter than RTU (make a drawing if you don't see this)
Instead of using the formula for RTU, round down PI from 3.14 to 3, and calculate:
(4*3) / 3 = 4
ONLY IF LOW ON TIME!



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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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abusaleh wrote: Bunuel, I am an avid follower of you. Would you pls draw the diagram according to the question and how you solved it. I haven't got it yet. Perimeter of circle \(= 2 \pi * 4 = 8\pi\) \(8\pi = Full Circle = 360\) What the angle for \(\frac{4\pi}{3}\) ??? Division factor \(= \frac{8\pi}{\frac{4\pi}{3}} = 6\) \(\frac{360}{6} = 60 = \angle RCU\) This makes\(\triangle RCU\) Equilateral which makes RU = Radius of circle = 4 Answer = D
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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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23 Oct 2016, 00:13
We know that length of an arc = Perimeter of a circle * ( Angle subtended at the center by the arc/360)
Perimeter can be found by = 2* Pi* R = 8 * Pi
Utilizing arc length formula Perimeter of a circle * ( Angle subtended at the center by the arc/360) = 4* Pi /3
=> 8 * Pi * ( Angle subtended at the center by the arc/360) = 4* Pi /3 =>Angle subtended at the center by the arc = 60 degrees
Now, end points at circle R & U will be equidistant from center of the circle = radius of circle , r = 4 We have a triangle where one angle is 60 and two sides equal . the triangle becomes an equilateral triangle.
Hence length of arc RU = 4
Option D



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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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23 Oct 2016, 01:58
use the ratio \(\frac{x}{360} = \frac{Arc RTU}{2*Pi*R}\) Then you find that x=60° Given that ORU is an isosceles triangle (0 is the centre of the circle), you have all angle = 60° therefore, the triangle is equilateral, the length of RU = R = 4



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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]
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23 Mar 2017, 04:14
sketch a circle and you will see that RTU length takes 1/6 of the whole circumference if the circle. 4P/3 divided by 8P the circle is 360 degrees and 1/6 of it is 60 degrees. as wenknow that R and U lie on the circle, the OR=OC. Based on the facts that ROC is 60*, OR=OU=4, RU=4 Hence the answer is D




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