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The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

\(area_{square}=\frac{d^2}{2}\) and \(area_{rhombus}=\frac{d_1*d_2}{2}\).

The difference is \(area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2\).

If \(x=1\), then the difference is 63;

If \(x=2\), then the difference is 252;

In order the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible.

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R? I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

IMO D !!

After reading question, just check for ans choices

no option with none, thus answer choice can take any value.

Area of square - area of rhombus \(Diagonal\)2 ...therefore if we consider scenario 1 as correct then only scenario which make sense is secnario 3 where value of secnario 1 is multiplied by 4...or we can say \(2\)2 thus...

Scenario 1 & Scenario 2 only make sense

D ...it was a calculated risk...a guess..i got it right

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

\(area_{square}=\frac{d^2}{2}\) and \(area_{rhombus}=\frac{d_1*d_2}{2}\).

The difference is \(area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2\).

If \(x=1\), then the difference is 63;

If \(x=2\), then the difference is 252;

In order the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible.

Answer: D

Hi Bunuel,

What make you take 15x as diagonal of square and the rest as diagonals of rhombus from the given ratio?

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

\(area_{square}=\frac{d^2}{2}\) and \(area_{rhombus}=\frac{d_1*d_2}{2}\).

The difference is \(area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2\).

If \(x=1\), then the difference is 63;

If \(x=2\), then the difference is 252;

In order the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible.

Answer: D

Hi Bunuel,

What make you take 15x as diagonal of square and the rest as diagonals of rhombus from the given ratio?

Not sure that I understand what you mean...

The diagonals of square are equal and the diagonals of rhombus are not. We are given the the ration of three diagonals is 15:11:9, hence \(d_s:d_1:d_2=15x:11x:9x\).
_________________

What makes you take 15x as diagonal of square and the rest as diagonals of rhombus from the given ratio?[/quote]

Not sure that I understand what you mean...

The diagonals of square are equal and the diagonals of rhombus are not. We are given the the ration of three diagonals is 15:11:9, hence \(d_s:d_1:d_2=15x:11x:9x\).[/quote]

My doubt is why you choose specifically 15x (among 15, 11 and 9) as diagonal of square. Why not d1 and d2 be 15x and 11x respectively, and Ds be 9x? If you consider this way, it could be "Area of rhombus - Area of square" since question stem mentions difference between area of square and area of rhombus. I want to know the logic behind your selection.

What makes you take 15x as diagonal of square and the rest as diagonals of rhombus from the given ratio?

Not sure that I understand what you mean...

The diagonals of square are equal and the diagonals of rhombus are not. We are given the the ration of three diagonals is 15:11:9, hence \(d_s:d_1:d_2=15x:11x:9x\).[/quote]

My doubt is why you choose specifically 15x (among 15, 11 and 9) as diagonal of square. Why not d1 and d2 be 15x and 11x respectively, and Ds be 9x? If you consider this way, it could be "Area of rhombus - Area of square" since question stem mentions difference between area of square and area of rhombus. I want to know the logic behind your selection.[/quote]

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively.
_________________

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

Hi Bunuel,

How do I know when I can just use ratios such as in this solution and assign some variable x (15x:11x:9x) and when I need to find LCM and calculate difference based on LCM solutions, e.g. LCM = 495, then proceed as you did, 495y/15, 495y/11, and 495y/9. Hope this is clear:)

Hii Bunuel Given that the ratio of the diagonal is ds:d1:d2=15x:11x:9xds:d1:d2=15x:11x:9x, for some positive integer xx (where dsds is the diagonal of square S and d1d1 and d2d2 are the diagonals of rhombus R).

areasquare=d22areasquare=d22 and arearhombus=d1∗d22arearhombus=d1∗d22.

The difference is areasquare−arearhombus=(15x)22−11x∗9x2=63x2areasquare−arearhombus=(15x)22−11x∗9x2=63x2.

If x=1x=1, then the difference is 63;

If x=2x=2, then the difference is 252;

In order the difference to be 126, xx should be 2√2, which is not possible.

Why is it not possible for x to be √2? Its just a number not necessarily an integer?

Hii Bunuel Given that the ratio of the diagonal is ds:d1:d2=15x:11x:9xds:d1:d2=15x:11x:9x, for some positive integer xx (where dsds is the diagonal of square S and d1d1 and d2d2 are the diagonals of rhombus R).

areasquare=d22areasquare=d22 and arearhombus=d1∗d22arearhombus=d1∗d22.

The difference is areasquare−arearhombus=(15x)22−11x∗9x2=63x2areasquare−arearhombus=(15x)22−11x∗9x2=63x2.

If x=1x=1, then the difference is 63;

If x=2x=2, then the difference is 252;

In order the difference to be 126, xx should be 2√2, which is not possible.

Why is it not possible for x to be √2? Its just a number not necessarily an integer?

Please read carefully: The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?
_________________

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

\(area_{square}=\frac{d^2}{2}\) and \(area_{rhombus}=\frac{d_1*d_2}{2}\).

The difference is \(area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2\).

If \(x=1\), then the difference is 63;

If \(x=2\), then the difference is 252;

In order the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible.

Answer: D

Why x= \(\sqrt{2}\) is not possible
_________________

You have to have the darkness for the dawn to come.

The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. \(63\)

II. \(126\)

III. \(252\)

A. \(I\) only B. \(II\) only C. \(III\) only D. \(I\) and \(III\) only E. \(I\), \(II\) and \(III\)

Given that the ratio of the diagonal is \(d_s:d_1:d_2=15x:11x:9x\), for some positive integer \(x\) (where \(d_s\) is the diagonal of square S and \(d_1\) and \(d_2\) are the diagonals of rhombus R).

\(area_{square}=\frac{d^2}{2}\) and \(area_{rhombus}=\frac{d_1*d_2}{2}\).

The difference is \(area_{square}-area_{rhombus}=\frac{(15x)^2}{2}-\frac{11x*9x}{2}=63x^2\).

If \(x=1\), then the difference is 63;

If \(x=2\), then the difference is 252;

In order the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible.

Answer: D

Why x= \(\sqrt{2}\) is not possible

Please pay attention to the highlighted part.
_________________