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The points R, T, and U lie on a circle that has radius 4 [#permalink]
 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 4*PI/3, what is the length of line segment RU? A. 4/3 B. 8/3 C. 3 D. 4 E. 6
Last edited by Bunuel on 27 Feb 2012, 09:27, edited 1 time in total.
Edited the question and added the OA
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Last edited by Bunuel on 05 Nov 2009, 12:03, edited 3 times in total.
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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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Bunuel - improved some math in your reply, but check that I did not mess it up accidentally.
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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=4bb 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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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]
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
(Give me a kudo if you like it)
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Re: PS Geometry (Circles/Arcs) [#permalink]
27 Oct 2010, 10:33
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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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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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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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Excellent, thanks for the confirmation Bunuel!
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The points R, T, and U [#permalink]
27 Feb 2012, 09:14
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 of line segment RU?
A. 4/3
B. 8/3
C. 3
D. 4
E. 6
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Re: The points R, T, and U [#permalink]
27 Feb 2012, 09:27
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Re: The points R, T, and U
[#permalink]
27 Feb 2012, 09:27
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