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# A circle with center O is inscribed in square WXYZ. Point P is the int

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Intern
Joined: 29 Mar 2013
Posts: 13
A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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16 Apr 2013, 15:41
2
8
00:00

Difficulty:

75% (hard)

Question Stats:

63% (02:51) correct 37% (03:08) wrong based on 155 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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Untitled.jpg [ 5.09 KiB | Viewed 3379 times ]
Manager
Joined: 27 Jul 2011
Posts: 145
Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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16 Apr 2013, 19:06
1
iwantto wrote:
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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Joined: 08 Dec 2012
Posts: 61
Location: United Kingdom
WE: Engineering (Consulting)
Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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29 Apr 2013, 13:50
3
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 $$0.64$$

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Joined: 08 Dec 2012
Posts: 61
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Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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29 Apr 2013, 13:58
sujit2k7 wrote:
iwantto wrote:
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

Intern
Status: Preparing...
Joined: 25 Mar 2013
Posts: 23
Location: United States
Sat: V
Concentration: Strategy, Technology
GMAT Date: 07-22-2013
GPA: 3.7
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Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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29 Apr 2013, 21:09
sujit2k7 wrote:
iwantto wrote:
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

Sujit,

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

Thanks
Current Student
Joined: 10 Jun 2015
Posts: 7
Location: India
GMAT 1: 720 Q49 V38
Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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05 Jul 2015, 05:55
1
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
Math Expert
Joined: 02 Sep 2009
Posts: 52385
Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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05 Jul 2015, 06:33
iwantto wrote:

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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Untitled.jpg

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Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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06 Jan 2016, 18:42
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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Joined: 03 Apr 2018
Posts: 3
Re: A circle with center O is inscribed in square WXYZ. Point P is the int  [#permalink]

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23 Apr 2018, 06:53
mvictor wrote:
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

I was able to deduce B in my head fairly quickly, and then when I saw your math, it confirmed my deduction. Thanks!
Re: A circle with center O is inscribed in square WXYZ. Point P is the int &nbs [#permalink] 23 Apr 2018, 06:53
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