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

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 153
Category: Geometry Circles; Triangles; Circumference
Page: 82
Difficulty: 600


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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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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

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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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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Answer = (D) 4

Perimeter of Circle \(= 8 \pi\)

Perimeter of arc \(= 4 \frac{\pi}{3}\)

So, \(8 \pi * \frac{1}{6} = 4 \frac{\pi}{3}\)

means Perimeter of arc is \(\frac{1}{6}\) th the Perimeter of Circle

So Angle ROU\(= \frac{360}{6} = 60\) Deg

Triangle ROU was already an isosceles triangle (as inscribed in circle) with equal sides = 4

Now as its a equilateral triangle, RU = 4
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Last edited by PareshGmat on 07 May 2014, 01:08, edited 1 time in total.

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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 25 Dec 2014, 01:45
I think the important point here is to understand the transition from isosceles triangle to equilateral. If you can quickly see that despite being isosceles, it is also equilateral triangle, you will be able to answer the Q. I missed this point and ended up marking wrong answer..

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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 09 Apr 2015, 05:58
So if the circumference of the circle is 8 (since radius 4). So 360 degrees is 8pi. (4pi/3)/8pi is 1/6. So RTU has at least one angle with 60 degrees, the other angles could be 30 + 90 or 60 + 60, or am I misintepreting the question?

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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 09 Apr 2015, 06:08
erikvm wrote:
So if the circumference of the circle is 8 (since radius 4). So 360 degrees is 8pi. (4pi/3)/8pi is 1/6. So RTU has at least one angle with 60 degrees, the other angles could be 30 + 90 or 60 + 60, or am I misintepreting the question?


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Angle RCU = 60°. \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\).
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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 11 Feb 2017, 13:01
Let consider O as the centre of circle and r as radius
2πr * ((∠ROU)/360) = 4π/3
After solving it ∠ROU = 60
now OR = OU (Radius of the circle)
which means ∠ORU = ∠OUR = 60 (Opposite angles of two opposite sides will be equal and some of three angles of triangle is 180)
as all three angles are 60 degree, which means it is equilateral triangle.
So Answer => OR = OU = RU = 4

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The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 11 Oct 2017, 11:39
Given radius is 4, you immediately know that triangle ROU is an Isosceles. But, since we are not given the degrees, we do not know at first if the triangle is an equilateral.

Circumference of a circle = 2pi r = 8pi
arc = x/360(circumference)

4pi/3/8pi = x/360
480/8pi = 60 = x

So, with 60 degrees as our angle in triangle ROU, and with two sides equal, we can deduce that the two unknown angles is 60, 60, making the triangle ROU an equilateral triangle.

All sides are equal in an equilateral triangle, therefore, RU = 4, Answer (D)

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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 16 Oct 2017, 15:55
Bunuel wrote:

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



We can let the center of the circle be C. Since the radius of circle C is 4, its circumference is 2πr = 2π4 = 8π.

We can use the following proportion to determine the central angle:

x/360 = (4π/3)/8π

x/360 = 4π/24π

x/360 = 1/6

x = 60

Now triangle RUC (i.e., the triangle formed by radii RC and UC and chord RU) is at least an isosceles triangle, since RC = UC = 4. However, since x = 60 degrees, or angle RCU = 60 degrees, triangle RUC must be equilateral because angles RUC and URC are also 60 degrees. Since triangle RUC is equilateral, RU = RC = 4.

Answer: D
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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 20 Oct 2017, 10:03
JeffTargetTestPrep wrote:
Bunuel wrote:

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



We can let the center of the circle be C. Since the radius of circle C is 4, its circumference is 2πr = 2π4 = 8π.

We can use the following proportion to determine the central angle:

x/360 = (4π/3)/8π

x/360 = 4π/24π

x/360 = 1/6

x = 60

Now triangle RUC (i.e., the triangle formed by radii RC and UC and chord RU) is at least an isosceles triangle, since RC = UC = 4. However, since x = 60 degrees, or angle RCU = 60 degrees, triangle RUC must be equilateral because angles RUC and URC are also 60 degrees. Since triangle RUC is equilateral, RU = RC = 4.

Answer: D


how do we know RC = UC?

Thanks

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The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 20 Oct 2017, 10:15
Length of arc RTU=4π/3
Angle subtended at the centre=\(\frac{4π}{3}\)/(2π*4)*360 =60 deg.
In triangle RCU, radius, RC=CU=4, hence each angle is 60 deg and triangle is equilateral triangle.
Hence RU=4
Answer D.
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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 19 Nov 2017, 00:51
Sorry this might seem like a silly question but why is a triangle formed here . I could find the degree measure of the and also find the length the circumference and am not understanding how to proceed further. Can someone suggest way to solve this question without assuming the triangle formed . thank you

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Re: The points R, T, and U lie on a circle that has radius 4. If [#permalink]

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New post 21 Nov 2017, 09:11
Why make a life harder and resolve via a triangle?
I got another comment - in this type of questions it is likely that a triangle is equilateral otherwise it would be too difficult to find the line segment, right? Wouldn't it be safe to assume so?

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Re: The points R, T, and U lie on a circle that has radius 4. If   [#permalink] 21 Nov 2017, 09:11
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