Question

In a $\Delta ABC$ median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.

Answer 1

Given : In $\Delta ABC,$ AD is median and AD is produced to X such that DX = AD

To prove : ABXC is a parallelogram

Construction : Join BX and CX

Proof : In $\Delta ABDand\Delta CDX$

AD= DX (Given)

BD = DC (D is mid points)

$\angle ADB=\angle CDX$ (Vertically opposite angles)

$\therefore \Delta ABD\cong \Delta CDX$ (SAS criterian)

$\therefore$ AB = CX (c.p.c.t)

and $\angle ABD=\angle DCX$

But these are alternate angles

$\therefore AB||CXandAB=CX$

$\therefore$ABXC is a parallelogram.