Question

Prove that :The diagonals of a rhombus bisect each other at right angle.

Answer 1

Let $ABCD$ is a rhombus.

$\Rightarrow$ $AB=BC=CD=DA$ [ Adjacent sides are eqaul in rhombus ]

In $△AOD$ and $△COD$

$\Rightarrow$ $OA=OC$ [ Diagonals of rhombus bisect each other ]

$\Rightarrow$ $OD=OD$ [ Common side ]

$\Rightarrow$ $AD=CD$

$\therefore$ $△AOD\cong △COD$ [ By SSS congruence rule ]

$\Rightarrow$ $\angle AOD=\angle COD$ [ CPCT ]

$\Rightarrow$ $\angle AOD+\angle COD={180}^o$ [ Linear pair ]

$\Rightarrow$ $2\angle AOD={180}^o.$

$\therefore$ $\angle AOD={90}^o.$

Hence, the diagonals of a rhombus bisect each other at right angle.