# Ex.8.1 Q3 Quadrilaterals Solution - NCERT Maths Class 9

## Question

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

## Text Solution

**What is known?**

The diagonals of a quadrilateral bisect each other at right angles.

**What is unknown?**

How we can show that it is a rhombus.

**Reasoning:**

To show that given quadrilateral is a rhombus, we have to show it is a parallelogram and all the sides are equal.

**Steps:**

Let be a quadrilateral, whose diagonals and bisect each other at right angle i.e

\(\mathrm{OA}\!=\!\mathrm{OC}, \mathrm{OB}\!=\!\mathrm{OD},\) and

\( \angle \mathrm{AOB}\!=\!\angle \mathrm{BOC}\!=\!\angle \mathrm{COD}\!=\!\angle \mathrm{AOD}\!=\!90^{\circ}\)

To prove \(ABCD\) a rhombus,

We have to prove \(ABCD\) is a parallelogram and all the sides of \(ABCD\) are equal.

In \(\Delta A O D \text { and } \Delta C O D,\)

\[\begin{align}O A&=O C\\(\text { Diagonals }&\text{bisect each other )} \\\\ \angle A O D&=\angle C O D \quad \text { (Given) } \\ O D&=O D \quad \text { (Common) } \\ ∴ \Delta A O D &\cong \Delta C O D \\ \text { (By SAS}&\text{ congruence rule) } \\\\ \therefore A D&=C D\quad\dots\text{(1)}\end{align}\]

Similarly, it can be proved that

\[\begin{align}A D=A B \text { and } C D=B C\quad\dots\text{(2)}\end{align}\]

From Equations (1) and (2),

\[\begin{align}A B=B C=C D=A D\end{align}\]

Since opposite sides of quadrilateral \(ABCD\) are equal, it can be said that \(ABCD\) is a parallelogram. Since all sides of a parallelogram \(ABCD\) are equal, it can be said that \(ABCD\) is a rhombus.