What is the circumference of the circle above?
(1) The length of arc XYZ is 18.
(2) r = s



I am completely stuck on this one. Can anyone please help?

thx, Khalid

statement 1:
to find the circumference of a circle from the length of an arc, you need to know what fraction of the circle the arc represents, or, equivalently, the number of degrees in the arc.
this statement provides the length of the arc, but no fraction or number of degrees; we don't know how much of the circle is occupied by the arc.
therefore, we can't determine the circumference.
insufficient.

statement 2:
if r = s, then s + 2s = 180, or s = 60. therefore, the triangle is equilateral. (alternatively, you don't have to do algebra; you can just recognize that the triangle is equilateral because all three of the angles have the same measure.)
this statement, though, provides no information about the size of ANYTHING.
insufficient.

together:
because the triangle is equilateral, r = 60.
the angle at Y is an inscribed angle, so the intercepted arc XZ measures 120 degrees.
this arc is therefore 1/3 of the circle.
the circle's circumference is therefore 3 x 18.
sufficient.

Isn't the circle's circumference 18*3\2=27?

why, yes, yes it is.

i had misread the arc in the problem as just "XZ", rather than "XYZ". since arc XZ is exactly half the length of arc XYZ, you're right - the circumference should be half the value i gave.

do remember, though, on data sufficiency problems such as this one, that the actual value of the circumference doesn't matter at all; all that matters is that you can find that value. therefore, neither of us should even be bothering to figure out the value; we're both wasting our time.

can u plz post a detailed explanation

of...?

this thread already contains a complete treatment of this problem.
see: second post from top.

that post contains one mistake, but that mistake has been corrected, thanks to the poster above.

so, i'm not sure what else you're looking for.
what else are you looking for?