Question

AD and BC are equal perpendiculars to a line segment AB. If AD and BC are on different sides of AB prove that CD bisects AB.

Answer 1

In ΔAOD and ΔBOC,
∠ AOD = ∠ BOC     ....(vertically opposite angles)
∠ DAO = ∠ CBO      ....(each 90°)
AD = BC                  ....(given)
∴ ΔAOD ≅ ΔBOC    ...(by AAS congruence criterion) 
⇒ AO = BO             ...(c.p.c.t.)
⇒ O is the mid-point of AB.
Hence, CD bisects AB.