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

In \(\triangle{AOD}\) and \(\triangle{BOC}\), we have,

Now, \(\angle{AOD}\) = \(\angle{BOC}\) ...(vertically opposite angles)

Also, \(\angle{DAO}\) = \(\angle{CBO}\) = \(90^\circ\)

and BD = BC ...(shown in the figure)

Therefore, \(\triangle{AOD}\) \(\displaystyle \cong \) \(\triangle{BOC}\) ...(SAS congruency test)

Hence, OA = OB ...(By CPCT)

Thus, we can say that, O is the mid-point of AB.

So, CD bisects AB.
Hence, proved