Question

If the diagonals of a quadrilateral bisect one another at right angles, then the quadrilateral is a:

• A. rhombus
• B. trapezium
• C. rectangle
• D. parallelogram

Answer 1

The correct option is A rhombus

If the diagonals of a quadrilateral bisect one another at right angles then the quadrilateral is a rhombus.
Also, if $ABCD$ is a quadrilateral and diagonals $AC$ and $BD$ bisect each other at at right angles,

then in $△AOB$ & $△AOD$,
$DO=OB$.....($O$ is mid point)
$AO=AO$.....(COMMON SIDE)
$\angle AOB=\angle AOD$ .....(Right angle)
Therefore, $△AOB\cong △AOD$ ...(By $RHS$ congruency criterion).

Then, $AB=AD$ (By $CPCT$ rule).

Similarly $AB=BC=CD=AD$ can be proved which means that quadrilateral is a rhombus.

Hence, option $A$ is correct.