Question

$AD$ and $BC$ are equal perpendiculars to a line segment $AB$ (see figure). Show that $CD$ bisects $AB$.

Answer 1

In $\Delta OBC$ and $\Delta OAD$
$\because \angle B=\angle A={90}^0$ (as $AD$ and $BC$ are perpendiculars)
and $OB=OA$ (given)
and $\angle BOC=\angle AOD$ (vertically opposite angles)
$\therefore \Delta OBC\cong \angle OAD$ (by AAS congruence rule)
$OA=OB$ (by c.p.c.t)
Similarly, $CD$ bisects $AB$
Hence proved.