Question

Prove that the four triangles formed by joining
in pairs the mid points of three sides of a triangle are congruent to each other

Answer 1

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Answer 2

GIVEN: A triangle ABC and D, E, F are the midpoints of the sides BC, CA and AB respectively.

TO PROVE: △AFE ≅△FBD ≅△EDC ≅△DEF

PROOF:

Since the segment joining the midpoints of the sides of a triangle is half of the third side. Therefore,

DE = 1/2AB => DE = AF = BF ____(i)

EF = 1/2BC => EF = BD = CD ____(ii)

DF = 1/2AC => DF = AE = EC ____(iii)

Now, in △DEF and △AFE, we have

DE = AF _____[From(i)]

DF = AE _____[From(ii)]

and, EF = FE _____[Common]

So, by SSS criterion of congruence, we obtain

△DEF ≅△AFE

Similarly, we have △DEF ≅△FBD and △DEF ≅△EDC

Hence, △AFE ≅△FBD ≅△EDC ≅△DEF.