Chapter 5.5, Problem 36WE

To determine

To Conclude: For the given qaudrilateral line segment PR and line segment SQ have same midpoint.

Explanation of Solution

Given information:

  

Points P,Q,R and S are the midpoints of the sides of quadrilateral ABCD.

The sides PR and SQ have the same midpoint, point O.

  

From the figure

Consider triangle ABC.

As P and Q are the midpoints of sides AB and BC respectively.

Therefore,

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

Also consider triangle ADC.

As R and S are the midpoints of CD and AD respectively.

Therefore,

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

From (1) and (2) equation

  $PQ=RS$ and $PQ\left|\right|RS$

This implies that PQRS is a parallelogram.

And in parallelogram each diagonal bisect each other.