Chapter 5.5, Problem 22WE

To determine

ToFind: the quadrilateral formed by joining mid-point of sides of a rectangle.

Explanation of Solution

Given

A rectangle is given.

Calculation:

Figure has been drawn as.

  

Using mid-point theorem.

  $\begin{array}{l}PQ\parallel AC\text{ and }PQ=\frac{1}{2}AC\\ SR\parallel AC\text{ and }SR=\frac{1}{2}AC\end{array}$

Using transitive property.

  $PQ\parallel SR$ therefore, PQRS is a parallelogram.

Since, ABCD is a rectangle.

  $\begin{array}{l}\therefore AD=BC\\ \frac{1}{2}AD=\frac{1}{2}BC\\ AS=BQ\end{array}$

In $\Delta APS\text{ and }\Delta BPQ$ .

  $\begin{array}{l}AP=BP\\ \angle PAS=\angle PBQ=90\\ AS=BQ\end{array}$

By SAS $\Delta APQ\cong \Delta BPQ$

  $\Rightarrow PS=PQ,\text{ By CPCT}$

Since, PQRS is a parallelogram with equal adjacent sides therefore it is a rhombus.

Therefore, parallelogramPQRS is a rhombus.