Question
$AD$ and $BC$ are equal perpendiculars to a line segment $AB.$ Show that $CD$ bisects $AB ($See figure$)$
✓
Answer
In $\triangle BOC$ and $\triangle AOD,$
$\angle OBC = \angle OAD = 90^\circ [$Given$]$
$ \angle BOC = \angle AOD [$Vertically Opposite angles$]$
$BC = AD [$Given$]$
$ \therefore \triangle BOC \cong \triangle AOD [$By $ASA$ congruency$]$
$ \Rightarrow OB = OA$ and $OC = OD [$By $C.P.C.T.]$
$\angle OBC = \angle OAD = 90^\circ [$Given$]$
$ \angle BOC = \angle AOD [$Vertically Opposite angles$]$
$BC = AD [$Given$]$
$ \therefore \triangle BOC \cong \triangle AOD [$By $ASA$ congruency$]$
$ \Rightarrow OB = OA$ and $OC = OD [$By $C.P.C.T.]$
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