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The points R, T and U lie on a circle that has radius 4. If the length : Problem Solving (PS)
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
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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: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
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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 the length [#permalink]
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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?


Check the image below:
Attachment:
Untitled.png
Untitled.png [ 3.98 KiB | Viewed 71491 times ]

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



I get that this appears in the Quantitative section, so we feel like there's some merit to doing math, but does GMAC give any fewer points for getting the right answer WITHOUT doing all that math? Nope. And it's faster. Please, for the love of HBS, get good at spotting opportunities to ballpark!!

RTU is 4(pi)/3. That's a smidge bigger than 4. Is RU longer than RTU, the same as RTU, or shorter than RTU? Shorter. Cool, E is wrong. Is RT a lot shorter than RTU or just a little shorter? Just a little. Great, A and B are definitely wrong. We are down to C and D. C is less than 75% of RTU and D is pretty close to RTU. Look back at the figure one more time. Do we want something as short as 3? Nope, C is wrong.

Answer choice D.

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

-------------------------------
Went through so many comments but still can't understand why RTU = 4 pi/3?
And how are the angles 60 degree?

I understand it's an isosceles triangle. Once we get RTU, we can find the angle then.

I'm stuck in this step!

Please help!

KarishmaB Bunuel BrentGMATPrepNow
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
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summerbummer 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

-------------------------------
Went through so many comments but still can't understand why RTU = 4 pi/3?
And how are the angles 60 degree?

I understand it's an isosceles triangle. Once we get RTU, we can find the angle then.

I'm stuck in this step!

Please help!

KarishmaB Bunuel BrentGMATPrepNow


summerbummer

The question tells us that RTU is 4pi/3.

As for why the angles are 60 degrees, we are told that the radius is 4. That means that the circumference is 8pi. 4pi/3 is 1/6th of 8pi, so the arc is 1/6th of the total circle. That means that the angle ROU is 1/6th of the number of degrees in the circle. There are 360 degrees, in a circle, so angle ROU is 60 degrees. The other two angles of triangle RTU are equal to each other and there are 120 degrees left in the triangle, so each of those other two angles must also be 60 degrees.
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
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summerbummer 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

-------------------------------
Went through so many comments but still can't understand why RTU = 4 pi/3?
And how are the angles 60 degree?

I understand it's an isosceles triangle. Once we get RTU, we can find the angle then.

I'm stuck in this step!

Please help!

KarishmaB Bunuel BrentGMATPrepNow


The circle has radius 4 so the circumference of the circle is \(2*\pi*4 = 8*\pi\)

The question tells us that the length of arc RTU is \(\frac{4*\pi}{3}\).

\(Length Of Arc = Q/360 * Circumference\) (where Q is the central angle subtended by the arc)

\(\frac{4*\pi}{3} = Q/360 * 8*\pi\)

We get Q = 60 degrees

So the central angle RCU = 60 degrees. So sum of angles CRU and RUC = 180 - 60 = 120 degrees

Since RC = UC = Radius of circle,
This means angels CRU and RUC are equal. Since their sum is 120, each of these is 60 degrees angle too. So triangle RCU has all angles of 60 degrees and is an equilateral triangle. Then RU = 4.

Answer (D)
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
Bunuel wrote:
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?


Check the image below:
Attachment:
Untitled.png

Angle RCU = 60°. \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\).


Bunuel
To follow up, how can we conclude that OU is equal to 4? What if OU is slightly longer e.g., 4.1? Thank you!
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
woohoo921 wrote:
Bunuel wrote:
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?


Check the image below:
Attachment:
Untitled.png

Angle RCU = 60°. \(RC=CU=r\) and \(RCU=CRU=CUR=60\) degrees. Hence \(RU=r=4\).



To follow up, how can we conclude that OU is equal to 4? What if OU is slightly longer e.g., 4.1? Thank you!



Triangle ORU is an isosceles, since 2 sides are the radius of the circle.
Angles opposite to the equal sides are also equal
From the arc we find the angle ROU=60 therefore the remaining angles are equal
Therefore It becomes a equilateral triangle. If radius =4 then, RU also equal to 4
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
Bunuel wrote:
SOLUTION

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 \(= 2\pi r=8\pi\).

\(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). So, the arc RTU is 1/6 th of the circumference. This means that \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle).

RCU is isosceles triangle as \(RC=CU=r\) and \(\angle RCU=\angle CRU=\angle CUR=60\) degrees. Hence \(RU=r=4\).

Answer: D.


Dumb question, why do you say that its an isoceles triangle?
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Re: The points R, T and U lie on a circle that has radius 4. If the length [#permalink]
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jpsot wrote:
Bunuel wrote:
SOLUTION

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 \(= 2\pi r=8\pi\).

\(\frac{RTU}{8*\pi}= \frac{(\frac{4*\pi}{3})}{8\pi}=\frac{1}{6}\). So, the arc RTU is 1/6 th of the circumference. This means that \(\angle{RCU}=\frac{360}{6}=60\) degrees (C center of the circle).

RCU is isosceles triangle as \(RC=CU=r\) and \(\angle RCU=\angle CRU=\angle CUR=60\) degrees. Hence \(RU=r=4\).

Answer: D.


Dumb question, why do you say that its an isoceles triangle?


RCU is an isosceles triangle because two of its sides, CR and CU, are radii of the circle and thus equal to each other.
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