Chapter 7.1, Problem 20PSC

To determine

To prove:The figure formed by joining midpoints of a quadrilateral is a parallelogram.

Explanation of Solution

Given information:

Quadrilateral and

mid points of quadrilateral joined in order.

Formula used:

Midpoint theorem of triangle

Proof:

Let ABCD be a parallelogram and P, Q, R and S are midpoints of AB, BC, CD and DA respectively

Join A and C

  

In $\Delta \text{ABC,}$ P and Q mid points of AB and BC respectively

  $\therefore \text{ PQ}\parallel \text{AC and PQ = }\frac{1}{2}\text{AC}$ ………..by midpoint theorem……..(i)

Again in $\Delta \text{DAC,}$ S and Rare midpoints of CD and DA respectively

  $\therefore \text{ SR}\parallel \text{AC and SR=}\frac{1}{2}\text{AC}$ ………by midpoint theorem……………..(ii)

From (i) and (ii), we have,

  $\text{PQ}\parallel \text{SR and PQ = SR}$

Thus, PQRS is a parallelogram