In the figure shown, two identical squares are inscribed in the rectan

Question

https://gmatclub.com/forum/download/file.php?id=18265 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? A. 8√2 B. 12 C. 12√2 D. 16 E. 18 Rectangle.png

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=18265

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?

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.

iLc · Answer

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

abhi398 · Answer

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

mbaiseasy · Answer

Answer is B. See Solution.

theGame001 · Answer

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

Thank You

Bunuel · Answer

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

theGame001 · Answer

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

Bunuel · Answer

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.

PareshGmat · Answer

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

mcelroytutoring · Answer

Attached is a visual that should help.

LogicGuru1 · Answer

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
 https://s31.postimg.org/6tl6fqprb/Rectangle.png

Sri07 · Answer

Hi Bunuel not understanding how we can assume that the dimensions of the rectangle have this relationship Length = 2*Width?

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

How do we know that the Width and the diagonal are the same ?

Bunuel · Answer

Check the diagram:

https://gmatclub.com/forum/download/file.php?id=18265

The width is equal to the diagonal of the squares and the length equal to two diagonals.

whichscore · Answer

Perimeter of rectangle = 2W + 2L = 18√2; Hence dividing the whole expression by 2 we get: W + L = 9√2. 
We also know that L = 2 diagonal of the squares and W = 1 diagonal of the squares. A square diagonal is given by S√2. Therefore, W + L = 1 S√2 + 2 S√2 --> 3 S√2 = 9√2 --> S = 3 --> Perimeter = 3 * 4 = 12.

Answer B.

RastogiSarthak99 · Answer

My method may be different than others:

when I see 2(l+b) = 18sqroot2

I noticed that 2 diagonals were equal to length and 1 diagonal was equal to breadth.

such that 2 sq root 2 * side = l and sq root * side = breadth

term side as "a"

and solve for a, which gives a = 3; perimeter of each square is 3*4 = 12

B

bumpbot · Answer

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In the figure shown, two identical squares are inscribed in the rectan

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