Geometry problem - Curcumference of a circle

This topic has expert replies
Legendary Member
Posts: 882
Joined: 20 Feb 2009
Thanked: 15 times
Followed by:1 members
GMAT Score:690
Can someone help me on the below geometry problem with detailed steps?

What is the circumference of the circle below (image attached)?
(1) The length of arc XYZ is 18.
(2) r = s
Attachments
Image.jpg

Junior | Next Rank: 30 Posts
Posts: 14
Joined: 04 Apr 2008

by sasi78 » Thu Apr 30, 2009 9:04 am
1. not sufficient --> we cannot use given arc lenght because it might vary for different angles
2. not sufficient -- > not sufficient data given to compute cir.

Both 1 & 2 sufficient -- > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,

2/3 * Circumference = 18 --> Cir = 27.

IMO, the answer is C.

What is the OA ?

Legendary Member
Posts: 882
Joined: 20 Feb 2009
Thanked: 15 times
Followed by:1 members
GMAT Score:690

by crackgmat007 » Thu Apr 30, 2009 9:10 am
I got it (I thought r as y, so got confused) anyways, thanks.

OA:C

Master | Next Rank: 500 Posts
Posts: 431
Joined: 10 Jan 2009
Thanked: 16 times
Followed by:1 members

by kanha81 » Thu Apr 30, 2009 1:31 pm
sasi78 wrote:1. not sufficient --> we cannot use given arc lenght because it might vary for different angles
2. not sufficient -- > not sufficient data given to compute cir.

Both 1 & 2 sufficient -- > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,

2/3 * Circumference = 18 --> Cir = 27.

IMO, the answer is C.

What is the OA ?
I am really flummoxed. Please help! How can r=s along with (i) help to solve the question. Isn't r radius and s the angle YXZ and YZX?

I am missing something... :(
Want to Beat GMAT.
Always do what you're afraid to do. Whoooop GMAT

Master | Next Rank: 500 Posts
Posts: 106
Joined: 04 Feb 2009
Location: Louisville, KY
Thanked: 8 times

by marcusking » Fri May 01, 2009 4:38 am
kanha81 wrote:
sasi78 wrote:1. not sufficient --> we cannot use given arc lenght because it might vary for different angles
2. not sufficient -- > not sufficient data given to compute cir.

Both 1 & 2 sufficient -- > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,

2/3 * Circumference = 18 --> Cir = 27.

IMO, the answer is C.

What is the OA ?
I am really flummoxed. Please help! How can r=s along with (i) help to solve the question. Isn't r radius and s the angle YXZ and YZX?

I am missing something... :(
I'm in the same boat I don't understand why it would mention anything about r. I thought the y was r, if it was then you would have enough info to solve the equation.

What does r stand for? Radius? Why would it give an angle measure equal to a length?

User avatar
Site Admin
Posts: 2567
Joined: 01 Jan 2009
Thanked: 712 times
Followed by:550 members
GMAT Score:770

by DanaJ » Fri May 01, 2009 5:05 am
The drawing misled you guys, since it's really small. That "y" you see on the picture is actually an "r" and it's just a random letter used to designate angle XYZ.

The notation is: r is the measurement of angle XYZ.

As you may imagine, you can't have r as radius, since it's impossible to say "length" = "measurement of angle" (it would be like comparing horses to chicken) in statement 2. r has to be an angle as well, since you only compare angles to angles.

Master | Next Rank: 500 Posts
Posts: 431
Joined: 10 Jan 2009
Thanked: 16 times
Followed by:1 members

by kanha81 » Fri May 01, 2009 9:53 pm
sasi78 wrote:1. not sufficient --> we cannot use given arc lenght because it might vary for different angles
2. not sufficient -- > not sufficient data given to compute cir.

Both 1 & 2 sufficient -- > If r=s, then it becomes an equilateral triangle and we can make another similar equilateral triangle with XYZ. As this is inscribed in circle, it divides the circle arc into 3 parts. Hence,

2/3 * Circumference = 18 --> Cir = 27.

IMO, the answer is C.

What is the OA ?
This might be a silly question, but "how" or "why" do you multiply the circumference by 2/3?

Isn't -
arc length = 18 = (n/360) * 2 * pi * r, where n=60
Want to Beat GMAT.
Always do what you're afraid to do. Whoooop GMAT

Legendary Member
Posts: 1035
Joined: 27 Aug 2008
Thanked: 104 times
Followed by:1 members

by scoobydooby » Fri May 01, 2009 10:17 pm
hey kanha18,

if you see the circle, it is made of three arcs-XY, YZ, XZ

statement 2 says r=s given that the base angles (s) are equal, that makes the triangle XYZ an equilateral triangle (easy to see if you join XZ, angle YXZ=s and angle YZX=s, r=s)

so all arcs would be equal in length=>XY=YZ=XZ

(XY+YZ+XZ=circumference)
so XY+XZ forms 2/3 of the circumference