Good question.
If the rectangle is not a square, it means that the if we equally divide the square into 2 rectangles, then their angles will not be 45 degrees (or they won't be 45-45-90 triangles).
However, they can be 30-60-90 triangles. The diagonal of this rectangle will pass through the origin of the circle and hence will have a measure of "2r". We know that in a 30-60-90 triangle, the ratio of the sides is 1:sqrt(3):2
So, if the hypotenuse is "2r", the sides will have be "1r" and sqrt(3)*r".
Therefore, the perimeter of this rectangle will be 2 * (r + sqrt(3)*r) = 2r(Sqrt(3) + 1) --> which is (B)

GMATGuruNY wrote:When a rectangle is inscribed in a circle, the DIAGONAL of the rectangle is also the DIAMETER of the circle.

I think what distracted me while solving this question was the statement, "if the rectangle is not square". I knew that the diagonal of a square will certainly be the diameter of the circle. But, now when I drawing inscribed rectangles in a circle, I notice that the diagonal of rectangle will certainly be the diameter of a circle.

GMAC often says that in a PS question, every bit of data is useful. Why is the data "if the rectangle is not square" given in this question?

Is it just a helper to say the triangles formed won't be a 45-45-90 triangle and to use the answer choices and go ahead presuming the triangle to be a 30-60-90 triangle? :roll:

Each diagonal of the rectangle is equal to the diameter of the circle. In addition Circle Theory says that a triangle in a semi-circle is right angled. Therefore the diameter is also the hypotenuse.
The hypotenuse is 2R and we need to find an answer that corresponds with a^2 + b^2 = (2R)^2 [by Pythagoras' Theorem], where and b are the lengths of the other 2 sides.
Then the perimeter will be P = a + b + 2R = 2R([a+b]/2R + 1)
Only one answer corresponds to such a pattern. That is ANSWER B = 2R(Sqrt 3 + 1)