Question

Assertion (A): The points $$A(1, -2), B(2, 3), C(-3, 2)$$ and $$D(-4, -3)$$ are the vertices of a parallelogram.

Reason (R): The midpoints of both the diagonals $$AC$$ and $$BD$$ are $$(-1, 0)$$.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The diagonals bisect each other at the same point, which is the criterion for a quadrilateral to be a parallelogram.

Step 2 – Reason: R is true:

$$M_{AC} = \left(\frac{1 + (-3)}{2}, \frac{-2 + 2}{2}\right) = (-1,0) $$ 

$$ M_{BD} = \left(\frac{2 + (-4)}{2}, \frac{3 + (-3)}{2}\right) = (-1,0)$$

Step 3 – Link: The diagonals share the same midpoint, so they bisect each other. This proves that ABCD is a parallelogram, and R correctly explains A.