Chapter 12, Problem 12CUR

To determine

To show: The diagonal are perpendicular to each other and it is a rhombus.

Explanation of Solution

Proof:

Draw a diagram of a parallelogram whose diagonals are perpendicular.

  

Figure (1)

Let $ABCD$ be a parallelogram. Diagonals AC and BD are perpendicular to each other.

In $\Delta ABM$ and $\Delta DMC$ ,

  $\begin{array}{c}\overline{DC}=\overline{AB}\left[\text{Opposite}\text{sides}\text{of}\text{parallelogram}\right]\\ \angle DMC=\angle AMB\left[\therefore \text{Both}90°\right]\\ \angle CDM=\angle MBA\left[\therefore \overline{DC}\parallel \overline{AB}\right]\end{array}$

Therefore, $\Delta ABM$ and $\Delta DMC$ are congruent to each other. So,

  $\begin{array}{c}DM=MB\\ CM=MA\end{array}$

Also, $AB=DC$ .

Both the diagonals of parallelogram are intersecting equally to each other at right angles. Hence, the diagonal are perpendicular to each other and it is a Rhombus.