Question

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

Answer 1

Its given that $AD=BC$....(i)

$AD$ is perpendicular to $AB$ i.e, $\angle OAD={90}^o$

$BC$ is perpendicular to $AB$ i.e, $\angle OBC={90}^o$

We have to prove that $CD$ bisects $AB$ i.e $OA=OB$

Since line $CD$ and $AB$ intersect,

$\angle AOD=\angle BOC$ [vertically opposite angle]....(ii)

In $△BOC$ and $△AOD$,

$\angle BOC=\angle AOD$ [from (ii)]

$\angle CBO=\angle DAO$ [both angles are ${90}^o$]

$\Rightarrow BC=AD$

$\therefore △BOC\cong △AOD$ [AAS Congruence Rule]

$\therefore BO=AO$ [CPCT]

Hence $CD$ bisects $AB.$