Question

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

Answer 1

Consider the triangles \( \triangle BOC \) and \( \triangle AOD \): 

\[ \angle BOC = \angle AOD \quad \text{(Since they are vertically opposite angles)} \]

\[ \angle CBO = \angle DAO \quad \text{(Both are right angles, 90º)} \]

\[ BC = AD \quad \text{(Given in the problem)} \]

Using the AAS congruence criterion (Angle-Angle-Side), we conclude that:

\[ \triangle BOC \cong \triangle AOD \]

From CPCT (Corresponding Parts of Congruent Triangles), we get:

\[ BO = AO \]

Therefore, the line \( CD \) divides the segment \( AB \) into two equal parts.