Question

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

Answer 1

Step-by-step explanation:

Given :

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

To prove :

△AFE≅△FBD≅△EDC≅△DEF.

Proof :

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

DE=

2

1

AB⟹DE=AF=BF ..........(1)

EF=

2

1

BC⟹EF=BD=CD .........(2)

DF=

2

1

AC⟹DF=AE=EC ..........(3)

Now, in △s DEF and AFE,

DE=AF

DF=AE

and, EF=FE

So, by SSS criterion of congruence,

△DEF≅△AFE

Similarly, △DEF≅△FBD and △DEF≅△EDC

Hence, △AFE≅△FBD≅△EDC≅△DEF

Hope this helps you!!