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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 \(RU=r=4\).

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

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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..

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

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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?

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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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

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

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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)

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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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.

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

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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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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

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

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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?