Problem Solving

In the figure shown above, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18 by squareroot2, then what is the perimeter of each square?
(A)8 by squareroot2
(B)12
(C)12 by squareroot2
(D)16
(E)18

Diagonal of each square is 18/(2*V2) or 9/V2

if side of the square is a diagonal will be aV2
so aV2 =9/V2 or a=9/2

hence perimeter of each square=4* 9/2 or 18

Ans option E

sorry, whats V2?

shouldn't it be?

let x = side of square

L of rectangle = 2 diagonals of square = 2xsqrt2
W of rectangle = 1 diagonal of square = xsqrt2
perimeter = 2(2xsqrt2) + 2(xsqrt2)

or

18sqrt2 = 4xsqrt2 + 2xsqrt2
18sqrt2 = 6xsqrt2

x = 3

perimeter of square = 4x or 4x3 = 12?

First of all 18 by sqrt(2) got me confused, I am going to take it as 18sqrt(2)

rectangle's perimeter = 2length + 2breadth ----(A)

from the figure we can see that length of rectangle = 2* diagonal of square (d)

from the figure we can see that breadth of rectangle = d


put these in (A)

we get 2(d+2d) = 6d

6d = 18*sqrt(2)
d= 3sqrt(2)

d = 2a^2 ('a' is the side of the square)

=> a^2 = 9*2 /2
a = 3

perimeter of square = 4*3 = 12 (ANSWER = B)

thanks a bunch!
i can't believe i didn't see that...sometimes studying doesn't pay.

jeffreydamian wrote:shouldn't it be?

let x = side of square

L of rectangle = 2 diagonals of square = 2xsqrt2
W of rectangle = 1 diagonal of square = xsqrt2
perimeter = 2(2xsqrt2) + 2(xsqrt2)

or

18sqrt2 = 4xsqrt2 + 2xsqrt2
18sqrt2 = 6xsqrt2

x = 3

perimeter of square = 4x or 4x3 = 12?

Sorry..I read the question as the breath of the rectangle in 18/V2 ...you are correct.It should be
6*aV2 =18/V2 [18 by squareroot2 and V2 is squareroot2]

in this way 4a=6..am I again making any mistake or the perimeter is 18 squareroot2? @san2009 can you please confirm.

I know this post is super old, but I tried it out today and others might too. Hope my approach - less algebraic more geometric - helps someone.

The square within the rectangle can be viewed as two special triangles put together, where the hypotenuse/diagonal of the square is root2.

The height of the rectangle is one hypotenuse long, and the length is 2 hypotenuses long, so the perimeter = 6 hypotenuses, all of which are factors of root2. The perimeter is 18root2. 18root2/6 = 3root2.

If the hypotenuse/diagonal of the square is 3*root2, then each side is 3*1 = 3 (ratios of special triangle 1:1:root2). Therefore each side = 3 and 3*4 = 12, the perimeter of the square.

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


In the figure shown above, two identical squares are inscribed in the rectangle. If the perimeter of the rectangle is 18 by squareroot2, then what is the perimeter of each square?
(A)8 by squareroot2
(B)12
(C)12 by squareroot2
(D)16
(E)18


==> Since the rectangle is consists of two big squares next to each other with each side of the square is a, then 6a=18sqrt2, a=3sqrt2. Then the diagonal side of the square is 6(right angled triangle with angles of 45 degrees has ratio of 1:1:sqrt2). Then each side of the small square inside has length of 3, and since the problem asks for the perimeter of the small square, 4*3 = 12, therefore the answer is B


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I am bit confused. I am new. The question says the parameter is 18 by squareroot2. However, the calculation shows parameter of the rectangle in 18 into squareroot2. confused.