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The points R, T, and U lie on a circle that has radius 4

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The points R, T, and U lie on a circle that has radius 4 [#permalink] New post 05 Nov 2009, 07:03
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 14 Nov 2013, 12:10, edited 2 times in total.
Edited the question and added the OA
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Re: circle- arc [#permalink] New post 05 Nov 2009, 07:26
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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.
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Last edited by Bunuel on 05 Nov 2009, 11:03, edited 3 times in total.
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Re: circle- arc [#permalink] New post 05 Nov 2009, 10:33
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] New post 05 Nov 2009, 10:54
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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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Re: circle- arc [#permalink] New post 05 Nov 2009, 11:34
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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] New post 13 Oct 2010, 09:53
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] New post 27 Oct 2010, 09:32
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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] New post 27 Oct 2010, 09:33
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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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Re: circle- arc [#permalink] New post 06 Dec 2010, 21:37
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] New post 07 Dec 2010, 02:05
Expert's post
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] New post 07 Dec 2010, 16:55
Excellent, thanks for the confirmation Bunuel!
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The points R, T, and U [#permalink] New post 27 Feb 2012, 08:14
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] New post 27 Feb 2012, 08:27
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Re: circle- arc [#permalink] New post 02 Oct 2013, 14:16
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] New post 03 Oct 2013, 00:28
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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: circle- arc   [#permalink] 03 Oct 2013, 00:28
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