Rhombus Properties and Diagonal Ratios

This question asks for the ratio of the lengths of the diagonals of a rhombus, given two statements about its side and diagonals. We need to determine if either statement alone, or both combined, is sufficient to answer the question.

Let's recall the properties of a rhombus:

• All four sides are equal in length. Let the side length be '\(s\)'.
• The diagonals bisect each other at right angles (90 degrees).
• Let the diagonals be '\(d_1\)' and '\(d_2\)'. When they bisect each other, they form four congruent right-angled triangles inside the rhombus. The sides of each right-angled triangle are '\(d_1/2\)', '\(d_2/2\)', and the hypotenuse is the side of the rhombus, '\(s\)'.

We can use the Pythagorean theorem for these right-angled triangles: \( \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2 \) \( \frac{d_1^2}{4} + \frac{d_2^2}{4} = s^2 \) \( d_1^2 + d_2^2 = 4s^2 \)

Analyzing Statement I

Statement I: One diagonal of the rhombus is equal to its side.

Let's assume \(d_1 = s\). We need to find the ratio \(d_1:d_2\).

Using the equation derived from the Pythagorean theorem:

\( d_1^2 + d_2^2 = 4s^2 \)

Substitute \(d_1 = s\): \( s^2 + d_2^2 = 4s^2 \) \( d_2^2 = 4s^2 - s^2 \) \( d_2^2 = 3s^2 \) \( d_2 = \sqrt{3s^2} \) \( d_2 = s\sqrt{3} \)

So, if \(d_1 = s\), then \(d_2 = s\sqrt{3}\). The ratio of the diagonals is \(d_1 : d_2 = s : s\sqrt{3} = 1 : \sqrt{3}\).

Therefore, Statement I alone is sufficient to determine the ratio of the lengths of the diagonals.

Analyzing Statement II

Statement II: The longer diagonal of the rhombus is equal to \(\sqrt{3}\) times its side.

Let's assume \(d_2\) is the longer diagonal, so \(d_2 = s\sqrt{3}\). We need to find the ratio \(d_1:d_2\).

Using the equation \(d_1^2 + d_2^2 = 4s^2\): \( d_1^2 + (s\sqrt{3})^2 = 4s^2 \) \( d_1^2 + 3s^2 = 4s^2 \) \( d_1^2 = 4s^2 - 3s^2 \) \( d_1^2 = s^2 \) \( d_1 = s \)

So, if the longer diagonal \(d_2 = s\sqrt{3}\), then the other diagonal \(d_1 = s\). The ratio of the diagonals is \(d_1 : d_2 = s : s\sqrt{3} = 1 : \sqrt{3}\).

Therefore, Statement II alone is sufficient to determine the ratio of the lengths of the diagonals.

Conclusion

Since both Statement I and Statement II independently provide enough information to calculate the ratio of the rhombus's diagonals (\(1 : \sqrt{3}\)), either statement alone is sufficient.

This corresponds to option (b).