The Length of the Diagonal of Square S as well as the Lengths of the Diagonals GMAT Problem Solving

Question: 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?

1. 63
2. 126
3. 252

1. I only
2. II only
3. III only
4. I and III only
5. I, II and III

Correct Answer: D

Solution and Explanation:

Approach Solution 1:
The problem statement states that:

Given:

• 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.

Find out:

• The difference between the area of square S and the area of rhombus R from the following options: I. 63 II. 126 III. 252

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).
Therefore, as per the formula of mathematics, we get:
\(Area_{square} = \frac{d^2}{2}\).and \(Area_{rhombus} = \frac{d_1* d_2}{2}\)

Therefore, \(Area_{square} = \frac{(15x)^2}{2}\) and \(Area_{rhombus} = \frac{11x * 9x}{2}.\)

Hence, the difference between the area of square and area of the rhombus is as follows:
=>\( Area_{square} − Area_{rhombus} = \frac{(15x)^2}{2}-\frac{11x * 9x}{2} = {63x}^2\)

If x = 1, then the difference is 63;
If x = 2, then the difference is 252;

In order to make the difference to be 126, x should be √2, which is not possible.
Hence only option I and III satisfies the question.

Approach Solution 2:

The problem statement states that:

Given:

• 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.

Find out:

• The difference between the area of square S and the area of rhombus R from the following options: I. 63 II. 126 III. 252

Let the diagonal of the square be d1 and the two diagonals of the rhombus are d2 and d3 respectively.

If the side of the square is s;
d1 = √2 * s
Squaring both sides, we get:
d1^2 = 2 s^2
=> s*s = d1*d½
=> Area of square = d1*d1/2

Area of rhombus in terms of diagonals = d2*d3/2

It is required to estimate the difference of area of square and rhombus. Therefore we get:
A = Area of the square - Area of the rhombus
A = d1*d1/2 - d2*d3/2
Given ratio, d1:d2:d3 = 15:11:9
Let common ratio be x, d1 = 15x, d2 = 11x, d3 = 9x

Therefore, A = 63x*x = 63x^2

Therefore, by analysing the options we get:

Option I: When A = 63, then we get:
=> 63x^2 = 63
=> x = 1 satisfies the condition since lengths of diagonals are integers, x has to be an integer.

Option II: When A = 126, then we get:
=> 63x^2 = 126
=> x^2 = 126/63
=> x^2 = 2
=> x = √2, which is not possible since lengths of diagonals are integers, x has to be an integer.
Hence this option gets eliminated.

Option III: When x = 252, then we get:
=> 63x^2 = 252
=> x^2 = 252/63
=> x^2 = 4
=> x = 2 satisfies the condition since lengths of diagonals are integers, x has to be an integer.

Hence only options I and III are possible and option II is not.

“The length of the diagonal of square S, as well as the lengths of the diagonals”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. To solve the GMAT Problem Solving questions, the candidates must possess sufficient knowledge regarding arithmetic, algebra and geometry. The candidates can go through several sorts of questions from the GMAT Quant practice papers that will enable them to improve their mathematical understanding.

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