Question: P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD. What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?<br>Statement-I: Area of the quadrilateral ABCD is 100 square unit.<br>Statement-II: Area of the quadrilateral PQRS is 50 square unit.

Consider the following for the next ten (10) items that follow : 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.

  1. A. If the Question can be answered by one of the Statements alone, but not by the other.
  2. B. If the Question can be answered by either Statement alone. ✓
  3. C. If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
  4. D. If the Question cannot be answered even by using both Statements together.

Correct Answer: B. If the Question can be answered by either Statement alone.

Explanation

According to Varignon's Theorem, the figure formed by joining the midpoints of any quadrilateral is a parallelogram, and its area is exactly half the area of the original quadrilateral. Thus, \text{Area}(PQRS) = \frac{1}{2} \text{Area}(ABCD). The difference in their areas is therefore simply equal to the area of PQRS. Statement-I directly gives the total area (100), making the difference 50. Statement-II directly gives the area of PQRS (50), which is precisely the difference. Hence, either statement alone is sufficient.

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