Question

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

Answer 1

Given:- $AD=BCAD\perp AB,\text{i.e.,}\angle OAD=90°BC\perp AB,\text{i.e.,}\angle OBC=90°$

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

Proof:- Since line $CD$ and $AB$ intersects.

$\angle AOD=\angle BOC\left(\text{Vertically opposite angles}\right)$

Now, in $△AOD$ and $△BOC$

$\angle AOD=\angle BOC$

$\angle CBO=\angle DAO(\because \text{Each}90°)$

$BC=AD\left(\text{Given}\right)$

By AAS congruency,

$△AOD\cong △BOC$

Therefore, by C.P.C.T.,

$AO=OB$

Hence proved.