Chapter 5.5, Problem 21WE

To determine

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

Explanation of Solution

Given

A rhombus is given.

Calculation:

Figure has been drawn as.

  

Using mid-point theorem.

  $\begin{array}{l}SP\parallel BD\text{ and }SP=\frac{1}{2}BD\\ RQ\parallel BD\text{ and }RQ=\frac{1}{2}BD\end{array}$

Using transitive property.

  $SP\parallel RQ$ therefore, PQRS is a parallelogram.

Since, diagonals of a rhombus bisect each other at right angles.

  $AC\perp BD$

Since, $SP\parallel BD,PQ\parallel AC\text{ and AC}\perp BD$

  $\begin{array}{l}\therefore SP\perp PQ\\ \Rightarrow \angle QPS={90}^{\circ }\end{array}$

Therefore, parallelogramPQRS is a rectangle.