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16 Apr 2013, 16:41
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B
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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
Attachment:
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16 Apr 2013, 20:06
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iwantto
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) .5
B) .64
C) .85
D) 1
E) 1.25
A) .5
B) .64
C) .85
D) 1
E) 1.25
From the above figure OX = \(\sqrt{2}\) and OP = 1
so Area of circle = \(pi*1*1= Pi\)
Diagonal XZ = \(2*\sqrt{2}\)
so side of square= 2
area = 4
diff in area = 4 - 3.14 = 0.85
IMO: C
29 Apr 2013, 14:50
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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
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
