Question

Prove mid point theorem

Answer 1

Midpoint theorem:- " If a line joining the midpoint of any two sides of a triangle is parallel to the third side and it's length is half of the third side."

Given:- In ∆ABC D and E are the midpoint of AB and AC respectively.

RTP:- DE||BC and DE = 1/2BC

Construction:- Draw a line CF such that CF||AD.

there is a continuation in the image.

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Answer 2

theorem  :  The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of it.

given :  ΔABC in which D and E are the midpoints of AB and BC respectively. Points D and E are joined.

to prove : DE║BC and DE =  1/2 BC

construction : Draw CF║BA, meeting DE produced to F.

proof : In ΔAED and ΔCEF, we have

           AE = CE  [∵ E is the midpoint of AC]

           ∠AED = ∠CEF  [ Vertically opposite angles]

           ∠DAE = ∠FCE   [ Alternate interior angles ]

        ∴ ΔAED ≅ ΔCEF [ By ASA congruence criterion ]

        ∴ AD = CF and DE = EF [ CPCT ]

        But , AD = BD

        ⇒ BD = CF and BD║CF

        ∴ BCFD is a parallelogram

        ⇒ DF║BC and DF = BC

         ⇒ DF║BC and DE = 1/2 DF = 1/2 BC  [∵ DE = EF ]

         Hence, DE║BC and DE = 1/2 BC

         Hence proved.