Question

$ABCD$ is a quadrilateral. $AM$ and $CN$ are perpendiculars to $BD,$ $AM=CN$ and diagonals $AC$ and $BD$ intersect at $O,$ then which one of the following is correct$?$

• A. $AO=OC$
• B. $BO=OD$
• C. $AO=BO$
• D. $CO=DO$

Answer 1

The correct option is A $AO=OC$

In $△AMO$ and $△CNO$
$AM=CN$ (Given)
$\angle AMO=\angle CNO={90}^{\circ }$
$\angle AOM=\angle CON$ (vert. opp. $\angle$s)
$\therefore$ $△AMO\cong △CNO$ (AAS)
$\Rightarrow$ $AO=OC$ (cpct)