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In the figure shown, two identical squares are inscribed in

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In the figure shown, two identical squares are inscribed in [#permalink]

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In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
Attachment:
Rectangle.png
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A. 8√2
B. 12
C. 12√2
D. 16
E. 18
[Reveal] Spoiler: OA

Last edited by Bunuel on 18 Sep 2012, 00:54, edited 1 time in total.
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Re: Geometry-Square within Rectangle [#permalink]

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udaymathapati wrote:
In the figure attached (refer file), two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
A. 8√2
B. 12
C. 12√2
D. 16
E. 18


The rectangle's \(width=d\) and \(length=2d\), where \(d\) is the diagonal of each square.

\(P_{rectangle}=2(d+2d)=18\sqrt{2}\) --> \(d=3\sqrt{2}\).

Now, \(d^2=s^2+s^2\), where \(s\) is the side of a square --> \(d^2=(3\sqrt{2})^2=18=2s^2\) --> \(s=3\) --> \(P_{square}=4s=12\).

Answer: B.
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Re: Geometry question [#permalink]

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dineesha wrote:
In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the
rectangle is 18‹\sqrt{2}, then what is the perimeter of each square?

A. 8\sqrt{2}
B. 12
C. 12‹\sqrt{2}
D. 16
E. 18


Please see figure in the attached file.


PERIMETER=2(A+B) WHERE A AND B ARE TWO SIDES OF THE RECTANGLE.....
A --> THE LENGTH
B-- > THE BREADTH

AS THE TWO SQUARES ARE IDENTICAL THE DIAGONALS ARE EQUAL TO B . THEREFORE A=2B ..

ON EQUATING WE WILL GET THE ANSWER

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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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Answer is B. See Solution.
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In the figure shown, two identical squares are inscribed in [#permalink]

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udaymathapati wrote:
In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
Attachment:
The attachment Rectangle.png is no longer available


A. 8√2
B. 12
C. 12√2
D. 16
E. 18


Given \(2l+2b=18√2\)
\(l+b=9√2\) {equation 1}

As seen in the diagram that length of the RECTANGLE is diagonal + diagonal OF SQUARE ; length = \(2d\)
As seen in the diagram that breadth of the RECTANGLE is diagonal of the SQUARE =\(d\)
As seen in the diagram the side of the square is \(x\)

Substituting these values in equation 1 gives us
\(2d+d=9√2\)
\(3d=9√2\)
\(d=3√2\) so the diagonal of the square is \(3√2\)
now \(side^2 + side^2 = diagonal ^2\) {simple pythagorus theorum}
\(x^2+x^2= (3√2)^2\)

\(2x^2= 9*2=18\)

\(x^2=\frac{18}{2} = 9\)

\(x=\sqrt{9}\)

\(x= 3\)the side of the square is 3 therefore its perimeter is 3*4=12

answer is B
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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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I solved it in a very easy way.
Lets take side of square is x. You can see from figure, two diagonals of squares = length of rectangle.
And one diagonal of square = width of rectangle.
So, as Length x Width = 36,
we can say (2 * root2x)* (root2x) = 36
x = 3
Perimeter of square = 12

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Re: Geometry-Square within Rectangle [#permalink]

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theGame001 wrote:
Bunuel wrote:

The length is twice the width, so if \(width=d\), then \(length=2d\).


This may sound a silly question but where is it stated that Length is twice the width? Is this a property of rectangle?


Not all rectangles have the ratio of width to length as 1 to 2.

From the figure we can see that the width equals to the diagonal of the inscribed square and the length equals to the two diagonals.
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Re: Geometry-Square within Rectangle [#permalink]

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New post 28 Aug 2010, 07:45
Hello :)

Let's name:

A width of the rectangle (the biggest line)
B height of the rectangle (the smallest line)
C width of the square

We know that 2 (A + B) = 18√2, so A + B = 9√2

We can also infer that A = 2B since A = 2 diagonal of the square and B = 1 diagonal of the square (see it on the figure to understand it more easily)

A = 3√2 and B = 6√2

From Pythagor, we have C² + C² = B²
<=> 2c² = (3√2)²
<=> 2c² = 9 * 2
<=> C = 3

So the perimeter of each square is 4 * 3 = 12

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Re: Geometry-Square within Rectangle [#permalink]

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New post 28 Aug 2010, 07:49
let each square is with side a & diagonal b. hence a = 1/\sqrt{2}b.
b is breadth of the bigger rectangle & 2b is the length of the rectangle.

perimeter of the rectangle is 2X(2b+b) = 6b = 18\sqrt{2}
b = 3\sqrt{2}

=> a = 3.
perimeter of each square = 12.

Answer is B
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Re: Geometry-Square within Rectangle [#permalink]

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New post 19 Apr 2011, 18:09
l+b = 9root(2) (l - length of rectange, b - breadth of rectangle)

Also, 2d + d = 9root(2) (d = Diagonal of square)

d = 3root(2)

Side of square = 3, so permieter = 4 * 3 = 12

Answer - B
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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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New post 26 Sep 2012, 01:40
Interesting questions and i like such questions.
Since diagonal of the square is equal to side of the square*sqrt2 then we have one side of the reqtangle is equal to two diagonal of the square and another side of the rectangle is equal to one diagonal. All the sides (perimiter) are equal to 6 diagonals. So the side of the square is equal to 18‹\sqrt{2}/6\sqrt{2}=3. Then perimiter of the square 3*4=12
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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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New post 21 Nov 2013, 14:45
udaymathapati wrote:
In the figure shown, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
Attachment:
Rectangle.png


A. 8√2
B. 12
C. 12√2
D. 16
E. 18


If y'all take a look you can tell that the length + width is equal to 3 diagonals of the square.
Therefore, Since 2(x+y) = 18 sqrt (2) then x+y = 9 sqrt (2)
Now as stated before we have 3s sqrt (2) = 9 sqrt (2)
s = 3, 's' stands for side of the square.
Perimeter = 12

Hope it helps
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Re: Geometry-Square within Rectangle [#permalink]

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New post 03 Jan 2014, 07:23
Bunuel wrote:
udaymathapati wrote:
In the figure attached (refer file), two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
A. 8√2
B. 12
C. 12√2
D. 16
E. 18


The rectangle's \(width=d\) and \(length=2d\), where \(d\) is the diagonal of each square.

\(P_{rectangle}=2(d+2d)=18\sqrt{2}\) --> \(d=3\sqrt{2}\).

Now, \(d^2=s^2+s^2\), where \(s\) is the side of a square --> \(d^2=(3\sqrt{2})^2=18=2s^2\) --> \(s=3\) --> \(P_{square}=4s=12\).

Answer: B.


Can I please ask why the width is D and length 2D?

Thank You

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Re: Geometry-Square within Rectangle [#permalink]

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New post 03 Jan 2014, 07:31
theGame001 wrote:
Bunuel wrote:
udaymathapati wrote:
In the figure attached (refer file), two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18√2, then what is the perimeter of each square?
A. 8√2
B. 12
C. 12√2
D. 16
E. 18


The rectangle's \(width=d\) and \(length=2d\), where \(d\) is the diagonal of each square.

\(P_{rectangle}=2(d+2d)=18\sqrt{2}\) --> \(d=3\sqrt{2}\).

Now, \(d^2=s^2+s^2\), where \(s\) is the side of a square --> \(d^2=(3\sqrt{2})^2=18=2s^2\) --> \(s=3\) --> \(P_{square}=4s=12\).

Answer: B.


Can I please ask why the width is D and length 2D?

Thank You


The length is twice the width, so if \(width=d\), then \(length=2d\).
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Re: Geometry-Square within Rectangle [#permalink]

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Bunuel wrote:

The length is twice the width, so if \(width=d\), then \(length=2d\).


This may sound a silly question but where is it stated that Length is twice the width? Is this a property of rectangle?

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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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New post 02 Sep 2014, 20:48
Perimeter of rectangle\(= 18\sqrt{2}\)

Lets say one side = x

other side \(= 9\sqrt{2} - x\)

When we divide the rectangle (as shown in fig), two squares would be formed

one side = x; other side \(= \frac{9\sqrt{2}}{2} - \frac{x}{2}\)

As square ABCD is formed, both sides should be equal

\(x = \frac{9\sqrt{2}}{2} - \frac{x}{2}\)

\(x = 3\sqrt{2}\)

Area of Square ABCD\(= 3\sqrt{2} * 3\sqrt{2} = 18\)

Area of inscribed square PQRS \(= \frac{1}{2} * 18 = 9\) (This is a thumb rule/property for inscribed square)

Length of a side of square PQRS \(= \sqrt{9} = 3\)

Perimeter of square PQRS= 3 * 4 = 12

Answer = B
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Re: In the figure shown, two identical squares are inscribed in [#permalink]

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New post 12 Apr 2016, 20:55
Attached is a visual that should help.
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Re: In the figure shown, two identical squares are inscribed in   [#permalink] 12 Apr 2016, 20:55
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