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 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.
Understanding the Geometry Problem
We are given a quadrilateral ABCD and P, Q, R, S are the mid-points of its sides AB, BC, CD, and DA, respectively. We need to find the difference between the area of quadrilateral ABCD and the area of the quadrilateral PQRS formed by joining these mid-points.
A key geometric property states that the quadrilateral formed by joining the mid-points of the sides of any quadrilateral is always a parallelogram, and its area is exactly half the area of the original quadrilateral.
Mathematically, let Area(ABCD) represent the area of quadrilateral ABCD and Area(PQRS) represent the area of quadrilateral PQRS. The property can be stated as:
\( \text{Area}(\text{PQRS}) = \frac{1}{2} \times \text{Area}(\text{ABCD}) \)
The question asks for the difference: Area(ABCD) - Area(PQRS).
Using the property above, we can express the difference in terms of Area(ABCD):
\( \text{Difference} = \text{Area}(\text{ABCD}) - \frac{1}{2} \times \text{Area}(\text{ABCD}) = \frac{1}{2} \times \text{Area}(\text{ABCD}) \)
Alternatively, we can express the difference in terms of Area(PQRS):
Since \( \text{Area}(\text{ABCD}) = 2 \times \text{Area}(\text{PQRS}) \),
\( \text{Difference} = \text{Area}(\text{ABCD}) - \text{Area}(\text{PQRS}) = 2 \times \text{Area}(\text{PQRS}) - \text{Area}(\text{PQRS}) = \text{Area}(\text{PQRS}) \)
So, the difference in area is equal to the area of the inner quadrilateral PQRS, and also equal to half the area of the outer quadrilateral ABCD.
Analyzing Statement Sufficiency
Now, let's analyze if the given statements are sufficient to answer the question "What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?".
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Statement-I: Area of the quadrilateral ABCD is 100 square unit.
Using the property derived above, the difference = \( \frac{1}{2} \times \text{Area}(\text{ABCD}) \).
Given Area(ABCD) = 100, the difference = \( \frac{1}{2} \times 100 = 50 \) square units.
Therefore, Statement-I alone is sufficient to answer the question.
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Statement-II: Area of the quadrilateral PQRS is 50 square unit.
Using the property derived above, the difference = Area(PQRS).
Given Area(PQRS) = 50, the difference = 50 square units.
Therefore, Statement-II alone is sufficient to answer the question.
Conclusion on Sufficiency
Since the question can be answered using Statement-I alone, and it can also be answered using Statement-II alone, the correct option is (b).
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.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm - 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 \(60^\circ\).
