Chapter 5.3, Problem 22WE

To determine

To prove: $PMNO$ is a parallelogram.

Explanation of Solution

Given:

  

A skew quadrilateral $SKEW$

  $M,N,O,$ and $P$ re the midpoints of $SK,KE,WE$ and $SW$ .

Proof:

In $\Delta SKW$ , as $M$ and $P$ are midpoints of $SK$ and $SW$ respectively,

  $\Rightarrow MP$ is parallel to $KW$ …(i)

and $MP=\frac{1}{2}KW$ …(ii)

In $\Delta EKW$ , as $N$ and $O$ are midpoints of $KE$ and $WE$ respectively,

  $\Rightarrow NO$ is parallel to $KW$ ...(iii)

and $NO=\frac{1}{2}KW$ …(iv)

Therefore, from (i) and (iii),

  $MP$ is parallel to $KW$ is parallel to $NO$ .

  $\Rightarrow MP$ is parallel to $NO$ ….(v).

From (ii) and (iv),

  $MP=\frac{1}{2}KW=ON$

  $\Rightarrow MP=ON$ ….(vi).

When opposite sides of a quadrilateral are congruent and parallel, the quadrilateral is a parallelogram.

From (v) and (vi),

  $PMNO$ is a parallelogram.

Hence proved.

Conclusion:

Therefore, $PMNO$ is a parallelogram.