Answer:
c. Statement-1 is true, Statement-2 is false
Step-by-step explanation:
Given:
- Assertion (A): In ΔABC, median AD is produced to X such that AD = DX. Then ABXC is a parallelogram.
- Reason (R): Diagonals AX and BC bisect each other at right angles.
The median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. In ΔABC, the median AD bisects side BC at point D. Therefore, according to the Segment Bisector Theorem, BD = DC.
As the median AD is extended to X such that AD = DX, and since we have already established that BD = DC, the diagonals AX and BC of quadrilateral ABXC bisect each other at point D.
Since the diagonals of quadrilateral ABXC bisect each other, this proves that quadrilateral ABXC is a parallelogram, confirming that the given assertion (A) is true.
The reason (R) given for ABXC being a parallelogram is false, as we cannot prove that the diagonals AX and BC bisect each other at right angles from the given information. Diagonals of a quadrilateral do not need to bisect each other at right angles to establish it as a parallelogram. The key proof lies in the fact that the diagonals bisect each other.
Additional Information
The quadrilaterals whose diagonals bisect each other at right angles are a square, a rhombus, and a kite. Only a square and a rhombus are considered parallelograms, since a kite with parallel sides is a rhombus.
