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

                                   

Answers (1)

In the given figure consider \DeltaAOD  and  \DeltaBOC.

(i)      AD  =  BC           (given)

(ii)   \angle A   =  \angle B          (given that the line AB is perpendicular to AD and BC)

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

Thus by AAS Postulate, we have   

                                                             \Delta AOD\ \cong \ \Delta BOC

Hence by c.p.c.t. we can write :                  AO\ =\ OB

And thus CD bisects AB.

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