Two squares in a rectangle - GMAT Prep 2.1

[rahulvsd]

Hi,

Please refer to the question in the image . I have a doubt in the explanation given. I agree that length of the rectangle here will be 2xsqrt2. How is the breadth of the rectangle xsqrt2? Is this got by the two 45:45:90 Triangles formed between the squares and the rectangle?

[Shalabh's Quants]

Hi,

Please refer to the question in the image . I have a doubt in the explanation given. I agree that length of the rectangle here will be 2xsqrt2. How is the breadth of the rectangle xsqrt2? Is this got by the two 45:45:90 Triangles formed between the squares and the rectangle?

Referring to the diagram...say side of the square is 'a', then diagonal of the square will be a.sqrt2.

Since length of the rectangle is 2 times diagonal, which will be equal to 2.a.sqrt2 &

width of the rectangle equals diagonal, which is equal to a.sqrt2.

Since, Perimeter of rectangle is 2(length+width)= 2.(2.a.sqrt2 + a.sqrt2 = 18.sqrt2.

This gives a = 3. or perimeter of square = 4a = 4.3 = 12.

[sam2304]

Please post larger images Its too small to view and zooming it just smudges the image.

2(l+b) = 18sqrt2
l+b = 9sqrt2
3b = 9sqrt2 [diagonals of the squares are same and both the squares are identical so two diagonals end to end forms the length and one diagonal forms the breadth so l = 2b]
b = 3sqrt2
side of the sq = 3 [we know that if the side of a square is a, then diagonal = a*sqrt2. Its the vice versa here]
perimeter of sq = 12

[rahulvsd]

I agree that two diagonals of the square form the length of the rectangle, how does one diagonal form the breadth of the rectangle? Are you saying this pictorially?

[GMATGuruNY]

Every triangle in the figure is a 45-45-90 triangle.
The sides of a 45-45-90 triangle in a ratio of x : x : x√2.

We can plug in the answers, which represent the perimeter of each square.
When the correct answer is plugged in, the perimeter of the rectangle will be 18√2.
The correct answer is probably a multiple of 4.

Answer choice B: 12
Thus, in each square, s=3.
The result is the following figure:



The 4 corner triangles:
Since x√2 = 3, x = 3/√2.
Within the perimeter, there are 8 lengths of 3/√2:
8(3/√2) = 24/√2 = (24√2)/(√2*√2) = 12√2.

The 2 middle triangles:
Since x = 3, x√2 = 3√2.
Within the perimeter, there are 2 lengths of 3√2:
2(3√2) = 6√2.

Perimeter of the rectangle = 12√2 + 6√2 = 18√2.
Success!

The correct answer is B.

[sam2304]

I agree that two diagonals of the square form the length of the rectangle, how does one diagonal form the breadth of the rectangle? Are you saying this pictorially?

I don't get your doubt mate . If you can agree with the two diagonals being the length of the rectangle, why can't the single one form the breadth of the rectangle ? If the squares are inscribed then all the vertices should touch the rectangle so the rectangle cannot be wider than the square's diagonal length. And since both diagonals are equal in length and perpendicular to each other, two diagonals end to end form the length and the one perpendicular to it is the breadth.

[rahulvsd]

Hi Sam2304,

Got it now. My Bad did not get it earlier.

Thanks mate!