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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]  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]  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 $$RU=r=4$$.

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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]  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

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Re: circle- arc [#permalink]  05 Nov 2009, 10:54
Expert's post
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]  05 Nov 2009, 11:34
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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

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]  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)

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Re: PS Geometry (Circles/Arcs) [#permalink]  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

(Give me a kudo if you like it)
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Re: PS Geometry (Circles/Arcs) [#permalink]  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]  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 $$RU=r=4$$.

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]  07 Dec 2010, 02:05
Expert's post
1
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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 $$RU=r=4$$.

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]  07 Dec 2010, 16:55
Excellent, thanks for the confirmation Bunuel!
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Re: circle- arc [#permalink]  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 $$RU=r=4$$.

Can you give me the geometric logic of finding angle RCU (360/6)
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Re: circle- arc [#permalink]  03 Oct 2013, 00:28
Expert's post
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 $$RU=r=4$$.

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: The points R, T, and U lie on a circle that has radius 4 [#permalink]  26 Jul 2014, 02:06
Bunuel, I am an avid follower of you. Would you pls draw the diagram according to the question and how you solved it. I haven't got it yet.
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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]  05 Aug 2014, 04:12
If you are running low on time - consider the following:

You know RU is a bit shorter than RTU (make a drawing if you don't see this)

Instead of using the formula for RTU, round down PI from 3.14 to 3, and calculate:

(4*3) / 3 = 4

ONLY IF LOW ON TIME!
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Re: The points R, T, and U lie on a circle that has radius 4 [#permalink]  06 Aug 2014, 01:12
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abusaleh wrote:
Bunuel, I am an avid follower of you. Would you pls draw the diagram according to the question and how you solved it. I haven't got it yet.

Perimeter of circle $$= 2 \pi * 4 = 8\pi$$

$$8\pi = Full Circle = 360$$

What the angle for $$\frac{4\pi}{3}$$ ???

Division factor $$= \frac{8\pi}{\frac{4\pi}{3}} = 6$$

$$\frac{360}{6} = 60 = \angle RCU$$

This makes$$\triangle RCU$$ Equilateral which makes RU = Radius of circle = 4

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Re: The points R, T, and U lie on a circle that has radius 4   [#permalink] 06 Aug 2014, 01:12
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