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Segment BDF bisects both square ABCD and square DEFG and has a length

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Bunuel
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Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   23 Aug 2018, 03:56
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Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36
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B
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 02:23
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Using the unkown multiplier approach:

according to the formula for 45-45-90 triangles, the side opposite to the 90° angle = \(x\sqrt{2}\) = 14\(\sqrt{2}\)
thus the length of the 2 triangles combined = 14

ratio of the sides of the 2 sqares: \(3x/4x-> total = 7x\)
7x = 14
x=2 = unkown multiplier

sides lengths are:
\(2*3 and 2*4\)

Areas of both squares \(= 6^2+8^2 = 100\)

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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 12:34
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hastee wrote:
Now, I have enough posts



Given in the question:-
Segment BDF has a length of \(14\sqrt{2}\) unit.
Here two cases can be concluded from the nomenclature of the segment BDF. viz.,
1) B - D - F
2) F - D - B

In succinct, D lies in between B and F or F and B.

According to your figure neither of the above terminology meets. Rather 'D' is at one corner making D_B_F or F_B_D. where B lies in between F and D.
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Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   Updated on: 23 Aug 2018, 06:25
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Bunuel wrote:
Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36


Let x and y are the length of the sides of the squares ABCD & DEFG respectively.

Given x:y=3k:4k
Given, segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units
Or, BD+DF=\(14\sqrt{2}\)
Or, \(\sqrt{2}(x+y)\)=\(14\sqrt{2}\)
Or, x+y=14
or, 3k+4k=14
Or, k=2

So, x=3k=6, y=4k=8

So sum of area of both the squares=\(6^2+8^2=100\)

Ans. (B)
Edit: Fixed the mathematical formula

Originally posted by PKN on 23 Aug 2018, 04:59.
Last edited by PKN on 23 Aug 2018, 06:25, edited 1 time in total.
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   23 Aug 2018, 06:17
PKN wrote:
Bunuel wrote:
Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36


Let x and y are the length of the sides of the squares ABCD & DEFG respectively.

Given x:y=3k:4k
Given, segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units
Or, BD+DF=\(14\sqrt{2}\)
Or, \sqrt{x}+[fraction]y[/fraction]=\(14\sqrt{2}\)
Or, x+y=14
or, 3k+4k=14
Or, k=2

So, x=3k=6, y=4k=8

So sum of area of both the squares=\(6^2+8^2=100\)

Ans. (B)
Hi PKN,
Can you please show the diagram of the question stem. I am not able to comprehend the question stem.

Thanks in Advance.
Sumit

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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   23 Aug 2018, 06:49
sumit411 wrote:
PKN wrote:
Bunuel wrote:
Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36


Let x and y are the length of the sides of the squares ABCD & DEFG respectively.

Given x:y=3k:4k
Given, segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units
Or, BD+DF=\(14\sqrt{2}\)
Or, \sqrt{x}+[fraction]y[/fraction]=\(14\sqrt{2}\)
Or, x+y=14
or, 3k+4k=14
Or, k=2

So, x=3k=6, y=4k=8

So sum of area of both the squares=\(6^2+8^2=100\)

Ans. (B)
Hi PKN,
Can you please show the diagram of the question stem. I am not able to comprehend the question stem.

Thanks in Advance.
Sumit

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app


Hi sumit411,

I have enclosed the required diagram.
Thank you.
Note:- Since x and y are of different ratio, hence squares are not identical though it seems identical in figure.
Attachments

Double square.JPG
Double square.JPG [ 18.6 KiB | Viewed 2134 times ]

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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   24 Aug 2018, 01:52
PKN wrote:
sumit411 wrote:
PKN wrote:
[quote="Bunuel"]Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36


Let x and y are the length of the sides of the squares ABCD & DEFG respectively.

Given x:y=3k:4k
Given, segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units
Or, BD+DF=\(14\sqrt{2}\)
Or, \sqrt{x}+[fraction]y[/fraction]=\(14\sqrt{2}\)
Or, x+y=14
or, 3k+4k=14
Or, k=2

So, x=3k=6, y=4k=8

So sum of area of both the squares=\(6^2+8^2=100\)

Ans. (B)
Hi PKN,
Can you please show the diagram of the question stem. I am not able to comprehend the question stem.

Thanks in Advance.
Sumit

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app


Hi sumit411,

I have enclosed the required diagram.
Thank you.
Note:- Since x and y are of different ratio, hence squares are not identical though it seems identical in figure.[/quote]Thanks for your effort :)
Surely helps.

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   26 Aug 2018, 19:23
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Bunuel wrote:
Segment BDF bisects both square ABCD and square DEFG and has a length of \(14\sqrt{2}\) units. If the ratio of the length of a side of ABCD to a side of DEFG is 3 : 4, what is the sum of the areas of the squares?


A. 196
B. 100
C. 64
D. 52
E. 36


We see that 14√2 is the sum of the lengths of the diagonal of square ABCD and the diagonal of square DEFG. Since the ratio of the sides of the two squares is 3:4, the ratio of the diagonals of the two squares is also 3:4. So we can let the diagonal of square ABCD = 3x and the diagonal of square DEFG = 4x and we have:

3x + 4x = 14√2

7x = 14√2

x = 2√2

Therefore, the diagonal of square ABCD is 3(2√2) = 6√2 and hence its side = 6 and its area = 6^2 = 36. Similarly, the diagonal of square DEFG is 4(2√2) = 8√2 and hence its side = 8 and its area = 6^2 = 64. So the sum of the areas of the two squares is 36 + 64 = 100.

Answer: B
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 10:42
How do we know that the two squares only intersect at one point (D)?

Given the conditions, is it possible that the two squares share point D and overlap at two sides? Thus the given diagonal 14(root2) would simply be the diagonal of the larger square?

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Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 11:28
hastee wrote:
How do we know that the two squares only intersect at one point (D)?

Given the conditions, is it possible that the two squares share point D and overlap at two sides? Thus the given diagonal 14(root2) would simply be the diagonal of the larger square?

Posted from my mobile device


Hi hastee,

Could you please enclose the diagram you have cited above?
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 12:01
I have too few posts to link an image.

My question: what verbiage in the question tells us that squares only intersect at point D. My concern is that 3 vertices of the smaller square may lie on the larger square, with the 4th point of the smaller square contained within the larger square.

The larger DEFG would contain, or intersect all points of square ABCD.
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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   27 Aug 2018, 12:02
Now, I have enough posts

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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   29 Aug 2018, 12:25
I made this diagram because someone mentioned they diagram already posted by someone else had the squares look the same size.

Though, you don't really have to go to all that of a hassle to solve this question.

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File comment: Diagram of squares with bisecting segment
IMG_20180830_002115.jpg
IMG_20180830_002115.jpg [ 4.36 MiB | Viewed 1780 times ]

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Re: Segment BDF bisects both square ABCD and square DEFG and has a length [#permalink] New post   20 Oct 2019, 18:02
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