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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.

Solution
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Given :

A triangle of ABC and D,E,F are the mid-points of sides BC,CA and AB respectively.

To prove :

AFEFBDEDCDEF.

Proof :

Since the segment joining the mid-points of the sides of a triangle is half of the third side. Therefore,

DE=12ABDE=AF=BF ..........(1)

EF=12BCEF=BD=CD .........(2)

DF=12ACDF=AE=EC ..........(3)

Now, in s DEF and AFE,

DE=AF

DF=AE

and, EF=FE

So, by SSS criterion of congruence,

DEFAFE

Similarly, DEFFBD and DEFEDC

Hence, AFEFBDEDCDEF

1333262_1111314_ans_30a557d37a3a48eb9b5a5f54d4d81c6e.png

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