A circle with center O is inscribed in square WXYZ. Point P is the intersection between the circle and the line connecting O with X. If \(PX =\sqrt{2} - 1\), what is the approximate area of the shaded parts of the square?

A. 0.5
B. 0.64
C. 0.85
D. 1
E. 1.25

Let side of the square be \(a\)
So, radius of the inscribed circle is \(\frac{a}{2} = OP\)

\(OX = \frac{1}{2}*\sqrt{2}a\)

\(PX = OX - OP\)

i.e \(\sqrt{2}-1 = \frac{1}{2}*\sqrt{2}a - \frac{a}{2}\)

i.e \(2*(\sqrt{2}-1) = a*(\sqrt{2}-1)\)

So \(a=2\); area of the square is 4.

Area of a circle inscribed in a square is \(\frac{pi}{4}*a^2\) i.e in this instance the area of the circle is \(pi\) (since \(a^2 = 4\))

The shaded portion is \(\frac{3}{4}*(4-pi) = \frac{3}{4}*0.86\) which is approximately \(0.64\)

Answer is B

Sujit,

I dont know how you got OX = \(\sqrt{2}\) and OP = 1. Could you break it down?

Also Nave is right question asks for shaded area and hence your answer needs to be multiplied by 3/4 or 0.75.

Thanks

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I was able to deduce B in my head fairly quickly, and then when I saw your math, it confirmed my deduction. Thanks!