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



Ans. We have ∠ABC = 90° and ∠BAD = 90°

          Also AB and CD intersect at O.

          ∴ Vertically opposite angles are equal.

          Now, in ΔOBC and ΔOAD, we have

                  ∠ABC = ∠BAD

[each = 90°]

                  BC = AD

[Given]

                  ∠BOC = ∠AOD

[vertically opposite angles]

          ∴ Using ASA criteria, we have

                  ΔOBC ≌ ΔOAD

          ⇒ OB = OA

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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