Question

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

Answer 1

See the diagram enclosed.

In the two triangles  AOD and BOC,
    Given AD ⊥ AB   and  BC ⊥ AB.
  So  AD ║ BC.
   So  ∠DAO = ∠CBO = 90°
   ∠AOD = ∠BOC  as vertically opposite angles or included angles.

   So the corresponding third angles in the triangles are also equal.
   ∠ODA = ∠OCB

   Given  AD = BC too.

   So  as per AAA property, both triangles are congruent.
   Hence  AO = OB.   CD bisects  AB..
   Proved.
  

Answer 2

Question :-

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

Answer :-

In ∆BOC and ∆AOD, we have

∠BOC = ∠AOD

BC = AD [Given]

∠BOC = ∠AOD [Vertically opposite angles]

∴ ∆OBC ≅ ∆OAD [By AAS congruency]

⇒ OB = OA [By C.P.C.T.]

i.e., O is the mid-point of AB.

Thus, CD bisects AB.

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