Question

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

Answer 1

Given

$AD$and $BC$ are equal perpendiculars to a line segment AB

To Prove

$CD$ bisects $AB$

Proof

Two triangles $\Delta AOD$ and $\Delta BOC$

$\angle AOD=\angle BOC$ {Vertically opposite angles}………………. (i)

$\angle OAD=\angle OBC$ {Given that they are perpendiculars}…. (ii)

$AD=BC$ {As given}………………………………………………… (iii)

From above three equation both the triangle satisfies “$AAS$” congruency criterion

So, $\Delta AOD\cong \Delta BOC$

$AO$ and $BO$ will be equal as they are corresponding parts of congruent triangles($CPCT$).

So, $AO=BO$

Hence, $CD$ bisects$AB$ at $O$.