Question

Suppose $X$, $Y$ and $Z$ are the midpoints of the sides $PQ$, $QR$ and $RP$ respectively of a triangle $PQR$. Prove that $XYRZ$ is a parallelogram.

Answer 1

Given: $x,y$ and $z$ are the mid-point of the sides $PQ,QR$ and $RP$ of $\Delta QPR,xy$ and $xz$ are joined.

To Prove: $XYRZ$ is a parallelogram.

Proof:

In $\Delta PQR,x,y,$ and $z$ are the mid-point of $PQ,QR$ and $RP$ respectively.

Also,

$XY||PR$ (mid-point theorem)

And,

$XY=\frac{1}{2}PR$ (mid-point theorem)

But we also know that,

$ZR=\frac{1}{2}PR$ (Given)

Therefore, $XYRZ$ is a Parallelogram (Since, One pair of opposite sides are parallel and equal)

Hence proved.