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rahulvsd: Master | Next Rank: 500 Posts; Posts: 184; Joined: Tue Sep 07, 2010 9:43 am; Thanked: 6 times; Followed by:1 members

Two squares in a rectangle - GMAT Prep 2.1

by rahulvsd » Wed May 16, 2012 9:17 am

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?

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Source: Beat The GMAT — Problem Solving | Wed May 16, 2012 9:17 am

Shalabh's Quants: Master | Next Rank: 500 Posts; Posts: 134; Joined: Fri Apr 06, 2012 3:11 am; Thanked: 35 times; Followed by:5 members

by Shalabh's Quants » Wed May 16, 2012 9:36 am

rahulvsd wrote: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.

Shalabh Jain,
e-GMAT Instructor

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sam2304: Legendary Member; Posts: 1239; Joined: Tue Apr 26, 2011 6:25 am; Thanked: 233 times; Followed by:26 members; GMAT Score:680

by sam2304 » Wed May 16, 2012 9:40 am

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

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rahulvsd: Master | Next Rank: 500 Posts; Posts: 184; Joined: Tue Sep 07, 2010 9:43 am; Thanked: 6 times; Followed by:1 members

by rahulvsd » Wed May 16, 2012 9:54 am

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?

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by GMATGuruNY » Wed May 16, 2012 12:09 pm

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 : xâˆš2.

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 18âˆš2.
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 xâˆš2 = 3, x = 3/âˆš2.
Within the perimeter, there are 8 lengths of 3/âˆš2:
8(3/âˆš2) = 24/âˆš2 = (24âˆš2)/(âˆš2*âˆš2) = 12âˆš2.

The 2 middle triangles:
Since x = 3, xâˆš2 = 3âˆš2.
Within the perimeter, there are 2 lengths of 3âˆš2:
2(3âˆš2) = 6âˆš2.

Perimeter of the rectangle = 12âˆš2 + 6âˆš2 = 18âˆš2.
Success!

The correct answer is B.

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sam2304: Legendary Member; Posts: 1239; Joined: Tue Apr 26, 2011 6:25 am; Thanked: 233 times; Followed by:26 members; GMAT Score:680

by sam2304 » Wed May 16, 2012 8:45 pm

rahulvsd wrote: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.