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705-805 (Hard)|   Geometry|      
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iwantto
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A circle with center O is inscribed in square WXYZ. Point P is the int 16 Apr 2013, 16:41
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B
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75% (hard)

Question Stats:

59% (02:47) correct 41% (03:00) wrong based on 109 sessions

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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

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B
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BukrsGmat
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A circle with center O is inscribed in square WXYZ. Point P is the int 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
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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
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A circle with center O is inscribed in square WXYZ. Point P is the int 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
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A circle with center O is inscribed in square WXYZ. Point P is the int 29 Apr 2013, 14:58
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sujit2k7
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

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
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Please note that it is shaded only on 3 sides
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A circle with center O is inscribed in square WXYZ. Point P is the int 29 Apr 2013, 22:09
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sujit2k7
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

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
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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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A circle with center O is inscribed in square WXYZ. Point P is the int 05 Jul 2015, 06:55
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1. OX is half of the diagonal of the square
2. radius of the circle is half of the side of the square
(sqrt(2)*a)/2 - a/2 = sqrt(2) - 1

this gives a = 2
and r = 1

Area of the square = 4

area of the circle = pi

Area of shaded portion = (Area of square - area of circle) *3/4

=.86*3/4 = .64
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A circle with center O is inscribed in square WXYZ. Point P is the int 05 Jul 2015, 07:33
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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. 0.5
B. 0.64
C. 0.85
D. 1
E. 1.25

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A circle with center O is inscribed in square WXYZ. Point P is the int 06 Jan 2016, 19:42
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tricky one.
so, we know PX. it can mean that OP=1 and OX=sqrt(2) or the diagonal of the square = 2(sqrt2). side of the square = 2. total area =4.
area of the circle = pi.

area of all 4 regions is ~ 4-pi or 3.14 => 0.86. now this is for all 4, we need to find area for 3 only:
0.86*3/4 ~ 0.64
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A circle with center O is inscribed in square WXYZ. Point P is the int 23 Apr 2018, 07:53
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mvictor
tricky one.
so, we know PX. it can mean that OP=1 and OX=sqrt(2) or the diagonal of the square = 2(sqrt2). side of the square = 2. total area =4.
area of the circle = pi.

area of all 4 regions is ~ 4-pi or 3.14 => 0.86. now this is for all 4, we need to find area for 3 only:
0.86*3/4 ~ 0.64
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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!
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A circle with center O is inscribed in square WXYZ. Point P is the int 10 Mar 2023, 10:04
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