AD and BC are equal perpendiculars to a line segment AB (see Fig. 7.18). Show that CD bisects AB.

Given: AD ⊥ AB, BC ⊥ AB, and AD = BC

To Prove: CD bisects AB or OA = OB

Proof: Consider two triangles △ BOC and △ AOD,

In △ BOC and △ AOD,

∠BOC = ∠AOD (Vertically opposite angles)

∠CBO = ∠DAO (Each 90º, since AD and BC are ⊥ to AB)

BC = AD (Given)

∴ △ BOC ≅ △ AOD (AAS congruence rule)

∴ BO = AO (By CPCT)

Thus, CD bisects AB and O is the mid-point of AB.

Hence, proved.