Bunuel

ABCD is a square inscribed in a circle and arc ADC has a length of \(\pi\sqrt{x}\). If a dart is thrown and lands somewhere in the circle, what is the probability that it will not fall within the inscribed square? (Assume that the point in the circle where the dart lands is completely random.)
(A) \(2x\)
(B) \(π(x) - 2x\)
(C) \(π(x) - \sqrt{2}(x)\)
(D) \(1 - \frac{2}{π}\)
(E) \(1 - \frac{2}{x}\)
Attachment:
2014-10-28_2033.png
The thing to note here is that since the square is inscribed in the circle, its diagonal will be the circle's diameter. Thereafter, the question becomes quite simple since getting any one dimension of these symmetrical figures is enough to get all others.
See:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2013/0 ... n-circles/Now if AC is the diagonal and the diameter, it means ADC is a semi circle.
Arc ADC = \(\pi*r\) which means \(r = \sqrt{x}\)
Area of circle = \(\pi*r^2 = \pi*\sqrt{x}^2 = \pi*x\)
Area of square = \(side^2 = (Diagonal/\sqrt{2})^2 = (2\sqrt{x}/\sqrt{2})^2 = 2x\)
Area of square/ Area of Circle \(= 2/\pi\)
Probability that it will land outside the square \(= 1 - 2/\pi\)
Answer (D)