Question

In the adjoining figure; P, Q and R are the mid-points of the sides BC, CA and AB, respectively, of ΔABC. Prove that ◻RQPB is a parallelogram.

Answer 1

Given: In triangle ABC, P, Q and R are the mid‑points of BC, CA and AB respectively.

To Prove: Quadrilateral RQPB is a parallelogram.

Proof (Step-wise):

1. R and Q are mid‑points of AB and AC respectively.

By the Midpoint Theorem the segment joining mid‑points RQ is parallel to BC and RQ = `1/2` × BC.

2. P is the mid‑point of BC, so PB is a part of the line BC.

Hence, PB is collinear with BC and therefore PB || RQ (From step 1).

3. Q and P are mid‑points of AC and BC respectively. By the Midpoint Theorem the segment QP is parallel to AB and QP = `1/2` × AB.

4. R is the mid‑point of AB, so BR is a part of the line AB.

Hence, BR is collinear with AB and therefore BR || QP (From step 3).

5. Since both pairs of opposite sides of quadrilateral RQPB are parallel (RQ || PB and QP || BR), RQPB is a parallelogram.

◻RQPB is a parallelogram.

Hence proved.