Question

In a quadrilateral \( ABCD \), \( M \) and \( N \) are the mid-points of the sides \( AB \) and \( CD \) respectively. If \( AD + BC = t \cdot MN \), then \( t = \)

• A. 4
• B. 2
• C. \( \frac{1}{2} \)
• D. \( \frac{3}{2} \)

Hint:

In geometry, the mid-line theorem is a powerful tool to relate the sides of a quadrilateral and its mid-line. Always remember that \( MN \) is half the sum of the opposite sides.

Answer 1

The Correct Option is B

Step 1: Understanding the geometry of the quadrilateral.
In any quadrilateral, if \( M \) and \( N \) are the mid-points of sides \( AB \) and \( CD \) respectively, the segment \( MN \) is known as the mid-line. The mid-line theorem states that the length of \( MN \) is half the length of the sum of the opposite sides, \( AD \) and \( BC \).

Step 2: Applying the mid-line theorem.
From the mid-line theorem, we know that: \[ MN = \frac{1}{2} (AD + BC) \] This gives us the equation \( AD + BC = 2 \cdot MN \), which means \( t = 2 \).

Step 3: Conclusion.
Thus, the correct value of \( t \) is 2, which makes option (B) the correct answer.