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

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

Answer is B. See Solution.

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 \(length=2d\).

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.

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

Attached is a visual that should help.

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

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 ?

Check the diagram:



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

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.

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

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