Question

Assertion : In $\Delta ABC$, median AD is produced to X such that AD = DX. Then, ABXC is a parallelogram.
Reason : Diagonals AX and BC bisect each other at right angles.
Which of the following is correct?

• A. If both assertion and reason are true and reason is the correct explanation of assertion
• B. If both assertion and reason are true but reason is not the correct explanation of assertion
• C. If assertion is true but reason is false
• D. If assertion is false but reason is true

Answer 1

The correct option is C If assertion is true but reason is false

In quadrilateral ABXC, we have,
AD = DX [Given]
BD = DC [AD is given as median]
$\angle ADB$ = $\angle XDC$ [Vertically Opposite Angles]
Thus, $\Delta ADB$ is congruent to $\Delta XCD$ by SAS congruency.

Hence, AB = XC [CPCT] (1)
$\angle BAD$ = $\angle CXD$ [CPCT]
$\angle BAD$ and $\angle CXD$ form a pair of alternate interior angles and since they are equal, AB||XC (2)
From (1) and (2), ABXC is proved to be a parallelogram, as we have proved that one pair of opposite sides are parallel and equal.
So, diagonals AX and BC bisect each other.
$\therefore$ ABXC is a parallelogram
$\therefore$ Assertion is true but reason is false.