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

 

Solution:

It is given that AD and BC are two equal perpendiculars to AB.

We will have to prove that CD is the bisector of AB

Now,

Triangles ΔAOD and ΔBOC are similar by AAS congruency since:

(i) ∠A = ∠B (They are perpendiculars)

(ii) AD = BC (As given in the question)

(iii) ∠AOD = ∠BOC (They are vertically opposite angles)

∴ ΔAOD ≅ ΔBOC.

So, AO = OB (by the rule of CPCT).

Thus, CD bisects AB (Hence proved).