Each item contains a Question followed by two Statements. Answer each item using the following instructions :
Choose option
(a) If the Question can be answered by one of the Statements alone, but not by the other.
(b) If the Question can be answered by either Statement alone.
(c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
(d) If the Question cannot be answered even by using both Statements together.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm
Analyzing the Tangential Quadrilateral Properties
The problem involves a quadrilateral \(ABCD\) where a circle touches all its four sides (\(AB\), \(BC\), \(CD\), \(DA\)). Such a quadrilateral is called a tangential quadrilateral.
A key property of a tangential quadrilateral is that the sums of its opposite sides are equal.
For quadrilateral \(ABCD\), this property means:
\(AB + CD = AD + BC\)The question asks for the perimeter of the quadrilateral. The perimeter (\(P\)) is the sum of all its sides:
\(P = AB + BC + CD + DA\)We can rewrite the perimeter formula using the sums of opposite sides:
\(P = (AB + CD) + (AD + BC)\)Evaluating Statement-I
Statement-I states that \(AB + DC = 10\) cm.
Because \(ABCD\) is a tangential quadrilateral, we know that the sum of opposite sides are equal:
\(AB + CD = AD + BC\)Using Statement-I, we know one pair of opposite sides sums to 10 cm:
\(AB + CD = 10 \text{ cm}\)Therefore, due to the property of tangential quadrilaterals, the other pair of opposite sides must also sum to 10 cm:
\(AD + BC = 10 \text{ cm}\)However, Statement-I alone only provides the sum for one pair of opposite sides (\(AB + CD\)). To find the perimeter, we need the sum of *both* pairs, which is \((AB + CD) + (AD + BC)\). While we know \(AB + CD = 10\), we deduced \(AD + BC = 10\) using the tangential property, but the statement itself doesn't directly provide the sum needed for the second half of the perimeter calculation independently.
Thus, Statement-I alone is not sufficient to determine the perimeter.
Evaluating Statement-II
Statement-II states that \(AD + BC = 10\) cm.
Again, because \(ABCD\) is a tangential quadrilateral:
\(AB + CD = AD + BC\)Using Statement-II, we know one pair of opposite sides sums to 10 cm:
\(AD + BC = 10 \text{ cm}\)Therefore, the other pair of opposite sides must also sum to 10 cm:
\(AB + CD = 10 \text{ cm}\)Similar to the analysis of Statement-I, Statement-II alone only gives the sum for one pair of opposite sides (\(AD + BC\)). The perimeter requires the sum of both pairs. While we know \(AD + BC = 10\), we deduced \(AB + CD = 10\) using the tangential property. The statement itself doesn't directly provide the sum needed for the first half of the perimeter calculation independently.
Thus, Statement-II alone is not sufficient to determine the perimeter.
Evaluating Both Statements Together
Let's consider both statements together:
- Statement-I: \(AB + DC = 10\) cm
- Statement-II: \(AD + BC = 10\) cm
Since the quadrilateral is tangential, we know that \(AB + CD = AD + BC\).
Statement-I tells us \(AB + CD = 10\) cm.
Statement-II tells us \(AD + BC = 10\) cm.
These two statements are consistent with the property of tangential quadrilaterals.
Now, let's calculate the perimeter \(P\) using both statements:
\(P = (AB + CD) + (AD + BC)\)Substituting the values from Statement-I and Statement-II:
\(P = (10 \text{ cm}) + (10 \text{ cm})\) \(P = 20 \text{ cm}\)Therefore, by using both statements together, we can successfully determine the perimeter of the quadrilateral.
Conclusion
The question "What is the perimeter of the quadrilateral?" can be answered using both Statement-I and Statement-II together, but neither statement alone is sufficient.
This corresponds to option (c).
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 \(60^\circ\). - P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD.
Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?
Statement-I: Area of the quadrilateral ABCD is 100 square unit.
Statement-II: Area of the quadrilateral PQRS is 50 square unit. - Question : What is the ratio of the lengths of diagonals of a rhombus?
Statement-I: One diagonal of the rhombus is equal to its side.
Statement-II: The longer diagonal of the rhombus is equal to \(\sqrt{3}\) times its side. - The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units.
Question : What are the values of r and R?
Statement-I: r is an integer.
Statement-II: R is an integer. - Let a, b, c and d be positive integers.
Question : Which one of a, b, c, d is closest to the product abcd?
Statement-I: \(a > b > c\)
Statement-II: c is not the smallest. - Let \(mn = k\), where m and n are prime numbers and k is an even number.
Question : What is the value of \(mn - n + 1\)?
Statement-I: \(m > n\)
Statement-II: One of the numbers is 2. - Question : If p is a positive integer, then what is the remainder when $p^n$ is divided by $p + 1$?
Statement-I: n is even.
Statement-II: p is even. Question : Is xy positive?
Statement-I: \(x = \sqrt[3]{-0.19683}\)
Statement-II: $y = \(\sqrt[3]{x} \)- Let a, b and c be the sides of a triangle ABC.
Question : Is the triangle equilateral?
Statement-I: \(a^2 + b^2 + c^2 = (ab + bc + ca)\)
Statement-II: \(3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca\) A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question : What is the integral value of \(k\) for which the expression \(4x^2-kx+1\)is positive?
Statement-I : \(k<-2\)
Statement-II: \(k>-4\)
Which one of the following is correctin respect of the above Question andthe Statements?
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 \(60^\circ\).
