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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

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 \(RU=r=4\).

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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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.

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.
_________________

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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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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