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:
In a quadrilateral ABCD, AB = 6 units, BC = 18 units, CD = 6 units, DA = 9 units. What is the length of diagonal BD?
Statement-I: The length of BD is an integer greater than 13.
Statement-II: The length of BD is an even integer.
Quadrilateral Diagonal Length Analysis
The question asks for the length of the diagonal BD in a quadrilateral ABCD, given the side lengths AB = 6, BC = 18, CD = 6, and DA = 9.
Geometric Constraints: Triangle Inequality Theorem
To find the possible length of the diagonal BD, we can consider the two triangles formed by the diagonal: triangle ABD and triangle BCD.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Applying this to our triangles:
Triangle ABD Analysis:
The sides are AB = 6, DA = 9, and BD.
- \(BD + AB > DA \Rightarrow BD + 6 > 9 \Rightarrow BD > 3\)
- \(BD + DA > AB \Rightarrow BD + 9 > 6 \Rightarrow BD > -3\) (This is always true since length is positive)
- \(AB + DA > BD \Rightarrow 6 + 9 > BD \Rightarrow 15 > BD\)
Combining these inequalities for triangle ABD, we get: \(3 < BD < 15\).
Triangle BCD Analysis:
The sides are BC = 18, CD = 6, and BD.
- \(BD + BC > CD \Rightarrow BD + 18 > 6 \Rightarrow BD > -12\) (Always true)
- \(BD + CD > BC \Rightarrow BD + 6 > 18 \Rightarrow BD > 12\)
- \(BC + CD > BD \Rightarrow 18 + 6 > BD \Rightarrow 24 > BD\)
Combining these inequalities for triangle BCD, we get: \(12 < BD < 24\).
Combining Constraints for Diagonal BD
To satisfy the conditions for both triangles, the length of BD must be within the intersection of the two ranges:
Intersection of \( (3 < BD < 15) \) and \( (12 < BD < 24) \) is \( 12 < BD < 15 \).
The possible lengths for BD must be strictly between 12 and 15.
Evaluating Statement I
Statement I says: "The length of BD is an integer greater than 13."
From our combined constraints, \(12 < BD < 15\). The integers in this range are 13 and 14.
If BD must be greater than 13, the only possibility from the combined range is BD = 14.
Conclusion for Statement I: This statement alone is sufficient to uniquely determine the length of BD as 14.
Evaluating Statement II
Statement II says: "The length of BD is an even integer."
From our combined constraints, \(12 < BD < 15\). The integers in this range are 13 and 14.
If BD must be an even integer, the only possibility from the combined range is BD = 14.
Conclusion for Statement II: This statement alone is sufficient to uniquely determine the length of BD as 14.
Final Conclusion
Since both Statement I and Statement II independently provide enough information to uniquely determine the length of the diagonal BD (which is 14 units), the question can be answered by using either statement alone.
Answer Justification
The question requires finding the length of diagonal BD. Based on the Triangle Inequality Theorem applied to triangles ABD and BCD, we derived the range \(12 < BD < 15\).
- Statement I restricts BD to integers greater than 13. Within the range \(12 < BD < 15\), the only integer greater than 13 is 14. Thus, Statement I is sufficient.
- Statement II restricts BD to be an even integer. Within the range \(12 < BD < 15\), the only even integer is 14. Thus, Statement II is sufficient.
Therefore, the question can be answered using either Statement I alone or Statement II alone.
Similar Questions
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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 \(60^\circ\).
