Question

$AD$ and $BC$ are equal perpendiculars to a line segment $AB$ in figure.$Showthat$CD$bisects$AB.$

Answer 1

Given : AD=BC

$AD\perp AB$ $\therefore \angle OAD={90}^{\circ }$

$BC\perp AB$ $\therefore \angle OBC={90}^{\circ }$

To prove : CD bisects AB i.e OA=OB

Proof : In $\Delta BOC$ and $\Delta AOD$

$\angle AOD=\angle BOC$...... (vertically opposite angles)

$\angle DAO=\angle CBO$ ...... (Both are ${90}^{\circ }$)

CB=AD ..... (given)

$\therefore \Delta BOC\cong \Delta \Delta AOD$ (AAS congruence rule)

$\therefore BO=AO$ ...... (corresponding sides of congruent triangles)