A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?
Question:
The diagonals of a rhombus ABCD are in the ratio 5:12. Is one of the diagonals equal to side of the rhombus?
Statement-I: The sum of the diagonals = 34 cm.
Statement-II: The length of a side = 13 cm.
Problem Setup: Rhombus Diagonals Ratio
The problem asks whether one of the diagonals of a rhombus is equal to its side, given that the diagonals are in the ratio 5:12. We need to determine if the provided statements are necessary to answer this question.
Rhombus Properties and Diagonal Relationships
Key properties of a rhombus relevant to this problem include:
- All four sides are equal in length.
- The diagonals bisect each other at right angles (90 degrees).
Consider a rhombus ABCD where the diagonals AC and BD intersect at point O. The intersection forms four congruent right-angled triangles (e.g., triangle AOB). The sides of triangle AOB are half the lengths of the diagonals (AO = AC/2, BO = BD/2) and the side of the rhombus (AB).
Let the lengths of the diagonals be \(d_1\) and \(d_2\). According to the Pythagorean theorem applied to triangle AOB:
\(s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2\)where s is the length of the side of the rhombus.
Diagonal Ratio Calculation
The question states that the diagonals are in the ratio 5:12. We can represent the lengths of the diagonals as \(d_1 = 5x\) and \(d_2 = 12x\), where x is a positive constant.
Now, let's substitute these into the Pythagorean theorem to find the side length s in terms of x:
\(s^2 = \left(\frac{5x}{2}\right)^2 + \left(\frac{12x}{2}\right)^2\) \(s^2 = \left(\frac{5x}{2}\right)^2 + (6x)^2\) \(s^2 = \frac{25x^2}{4} + 36x^2\)To add these terms, we find a common denominator:
\(s^2 = \frac{25x^2}{4} + \frac{144x^2}{4}\) \(s^2 = \frac{25x^2 + 144x^2}{4}\) \(s^2 = \frac{169x^2}{4}\)Taking the square root of both sides to find s:
\(s = \sqrt{\frac{169x^2}{4}}\) \(s = \frac{13x}{2}\)Diagonal vs. Side Equality Check
The question asks if one of the diagonals is equal to the side of the rhombus. We need to check if \(d_1 = s\) or \(d_2 = s\).
Case 1: Is \(d_1 = s\)?
Check if \(5x = \frac{13x}{2}\).
Multiplying both sides by 2 gives \(10x = 13x\).
Subtracting \(10x\) from both sides gives \(0 = 3x\), which implies \(x = 0\). A value of \(x=0\) would mean the diagonals have zero length, resulting in a degenerate rhombus (a point), which is not typically considered. For any actual rhombus (\(x > 0\)), \(d_1 \neq s\).
Case 2: Is \(d_2 = s\)?
Check if \(12x = \frac{13x}{2}\).
Multiplying both sides by 2 gives \(24x = 13x\).
Subtracting \(13x\) from both sides gives \(11x = 0\), which implies \(x = 0\). Again, this suggests a degenerate rhombus. For any actual rhombus (\(x > 0\)), \(d_2 \neq s\).
Therefore, based purely on the ratio of the diagonals being 5:12, we can definitively conclude that neither diagonal can be equal to the side of the rhombus.
Sufficiency of Question Information
Since the information provided within the question itself (the ratio of the diagonals being 5:12) is sufficient to answer the question "Is one of the diagonals equal to the side of the rhombus?", we do not need to use either Statement I or Statement II.
The analysis above demonstrates that the ratio 5:12 inherently prevents a diagonal from equaling the side length. This means the question can be answered independently of the additional information given in the statements.
Similar Questions
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\). - A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD.
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Question : Which one of a, b, c, d is closest to the product abcd?
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Important Questions from Data Sufficiency
- In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Harsh, Ketan, Mayank, Rohit, and Sidharth, each have different height. Who is the shortest?
(I) Mayank is taller than Harsh but shorter than Ketan.
(II) Rohit is taller than Harsh but shorter than Mayank. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Kunal, Mohit, Rohan, Sandeep, and Tarun - each have different age. Who is the oldest?
(I) Mohit is older than Kunal but younger than Sandeep.
(II) Tarun is younger than Sandeep but older than Mohit. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Question:
Among 5 trees, L, M, N, T and V each having a different height, which tree is the shortest?
Statements:
(I) N is shorter than only three trees.
(II) N is taller than M but shorter than T. Below is given a pair of questions, which are required to be answered, and a triad of statements. Which group of statements is sufficient to answer both the questions?
Question 1: Who is sitting opposite to Q?
Question 2: Who is sitting to the right of Q?
Statement 1 : P and U are sitting on left and right sides of T, respectively
Statement 2: P, Q, R, S, T and U are sitting around a round table with six chairs
Statement 3: Q is NOT sitting on the immediate left side of R
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\).
