Chapter 0.1, Problem 56E

To determine

To Show: The midpoint of consecutive sides of any quadrilateral are connected the result is parallelogram.

Explanation of Solution

Given information:

Mid points of consecutive sides of any quadrilateral are connected.

Calculation:

  

Let $ABCD$ is a quadrilateral and $P,Q,R,S$ are the mid points of the sides $AB,BC,CD,DA$ respectively. Now join $A$ and $C$ .It gives two triangles $\Delta ADC,\Delta ABC$ .

In $\Delta ADC$ ,

  $S$ is the mid-point of $AD$ and $R$ is the mid-point of $CD$ .

The mid-point theorem says that line segments joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.

Therefore $SR\left|\right|AC$ and $SR=\frac{1}{2}AC$ ....(1)

Now In $\Delta ABC$ ,

  $P$ is the mid-point of $AB$ and $Q$ is the mid-point of $BC$ .

  $PQ\left|\right|AC$ and $PQ=\frac{1}{2}AC$ ....(2)

From (1) and (2)

  $PQ=SR$ and $PQ\left|\right|SR$

So, in PQRS, one pair of opposite sides are parallel and equal. Hence PQRS is a parallelogram.

  $PR$ and $SQ$ are the diagonals of parallelogram $PQRS$ .

So, $OP=OR$ and $OQ=OS$ (Diagonals of a parallelogram bisect each other).

It is proved that the midpoint of consecutive sides of any quadrilateral are connected the result is parallelogram.