Geometry Arc

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kartik1979: Master | Next Rank: 500 Posts; Posts: 115; Joined: Thu Nov 27, 2008 3:17 am; Thanked: 1 times

Geometry Arc

by kartik1979 » Mon Mar 23, 2009 10:28 am

The points R, T, and U lie on a circle that has radius 4. If the length of arc RTU is 4p/3

what is the length of line segment RU?
A.43
B 8/3
C. 3
D. 4
E. 6

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Source: Beat The GMAT — Problem Solving | Mon Mar 23, 2009 10:28 am

kartik1979: Master | Next Rank: 500 Posts; Posts: 115; Joined: Thu Nov 27, 2008 3:17 am; Thanked: 1 times

by kartik1979 » Mon Mar 23, 2009 10:30 am

a is 4/3 not 43

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truplayer256: Master | Next Rank: 500 Posts; Posts: 392; Joined: Thu Jan 15, 2009 12:52 pm; Location: New Jersey; Thanked: 76 times

by truplayer256 » Mon Mar 23, 2009 12:09 pm

The answer is 4. Here's why:

We're given that arc RTU is 4pi/3 and that the radius is 4. From this information, we can assume that the diameter of the circle is 8 and use the formula for circumference to find out what degree measure makes arc RTU 4pi/3.

C=pi*diameter
C=8pi
x=degree measure
8pi*x/360=4pi/3
x=4pi/3*360/8pi=60 degrees

The length of RU intersects the two radii of the circle, which makes a triangle. Knowing that two sides of the triangle are going to be the same (because of the equal radii) and also knowing the angle RTU equals 60, we know that the triangle is equilateral. Since, RT=4 and TU=4, RU must also equal 4.

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KICKGMATASS123: Master | Next Rank: 500 Posts; Posts: 113; Joined: Thu Feb 26, 2009 8:13 am; Location: New Jersey; GMAT Score:650

by KICKGMATASS123 » Mon Mar 23, 2009 9:16 pm

it's hard to visualize why RU is 4.

I get how it forms an isosceles triangle consisting of 2 radii with a 60 degree angle.. however i was lost after that..

Can someone help please?

Thanks in advance

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kartik1979: Master | Next Rank: 500 Posts; Posts: 115; Joined: Thu Nov 27, 2008 3:17 am; Thanked: 1 times

by kartik1979 » Tue Mar 24, 2009 5:58 am

Answer given to said problme is - A

but they didnt explain how they arrived at it

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lilu: Master | Next Rank: 500 Posts; Posts: 148; Joined: Wed Dec 10, 2008 5:13 pm; Location: SF, CA; Thanked: 12 times

by lilu » Tue Mar 24, 2009 3:10 pm

We know that the arc is 1/6 of the circumference (4pi/3 / 2pi4 (the length of circ.)= 1/6), so the angle ROU (O-center)=60 degrees.
We know that the two sides of the triangle ROU=4, so it is an isolesce triangle and since the measure of the angle ROU is 60 degrees, we can determine that this triangle is equilateral. So we can conclude that the line segment RU = 4.

We need to draw a diagram in order to better understand what the problem is asking.