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16 Sep 2014, 01:29
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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 \(III\)
I. \(63\)
II. \(126\)
III. \(252\)
A. \(I\) only
B. \(II\) only
C. \(III\) only
D. \(I\) and \(III\) only
E. \(I\), \(II\) and \(III\)
16 Sep 2014, 01:29
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Official Solution:
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 diagonals 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).
The area of the square is given by \(area_{square}=\frac{d_s^2}{2}\), and the area of the rhombus is given by \(area_{rhombus}=\frac{d_1*d_2}{2}\).
The difference between the areas 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 for the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible since \(x\) must be an integer.
Answer: D
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 diagonals 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).
The area of the square is given by \(area_{square}=\frac{d_s^2}{2}\), and the area of the rhombus is given by \(area_{rhombus}=\frac{d_1*d_2}{2}\).
The difference between the areas 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 for the difference to be 126, \(x\) should be \(\sqrt{2}\), which is not possible since \(x\) must be an integer.
Answer: D
General Discussion
17 Aug 2017, 06:16
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This is really really a good sum
Forced me to think like crazy
Bunuel you are the best If i ever get a Q50 it will be because of you
Forced me to think like crazy
Bunuel you are the best If i ever get a Q50 it will be because of you
