In $\Delta PQR$ are the mid points of side AB, BC and AC respectively. PQ and QR are joined. Prove that BRQP is a parallelogram.

Given: P,Q,R are mid-points of AB, AC, BC.

To prove: BRQP is parallelogram.

Proof:

Statement

1. $PQ\parallel BC$

Reason-Mid-point theorem as P,Q are mid-points

2. $PQ\parallel \frac{1}{2}BC$

$\Rightarrow PQ=BR$

Reason: mid-point theorem as P,Q are mid-points

3. $QR\parallel AB$

Reason- R is mid-point of BC

4. $QR=\frac{1}{2}AB$

$\Rightarrow QR=PB$

Reason: mid-point theorem as Q,R are mid-points

P is mid-point of AB

5. BRQP is parallelogram

Reason: pairs of opposite sides are parallel and equal.

Hence proved.