Question

Assertion (A): The points $$A(0, -4), B(6, 2), C(3, 5)$$ and $$D(-3, -1)$$ taken in that order are the vertices of a rectangle $$ABCD$$.

Reason (R): The midpoint of both $$AC$$ and $$BD$$ is $$\left(\frac{3}{2}, \frac{1}{2}\right)$$.

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 but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The adjacent sides have slopes $$1$$ and $$-1$$, so they are perpendicular; the quadrilateral is a rectangle.

Step 2 – Reason: R is true. The midpoints of both diagonals are $$\left(\frac{3}{2}, \frac{1}{2}\right)$$.

Step 3 – Link: Equal diagonal midpoints show that the diagonals bisect each other, proving the quadrilateral is a parallelogram. This alone does not establish that it is a rectangle; perpendicular adjacent sides are also needed. Thus, R does not fully explain A.