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

Given :

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

To prove :

$△AFE\cong △FBD\cong △EDC\cong △DEF$.

Proof :

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

$DE=\frac{1}{2}AB⟹DE=AF=BF$ ..........(1)

$EF=\frac{1}{2}BC⟹EF=BD=CD$ .........(2)

$DF=\frac{1}{2}AC⟹DF=AE=EC$ ..........(3)

Now, in $△s$ $DEF$ and $AFE,$

$DE=AF$

$DF=AE$

and, $EF=FE$

So, by SSS criterion of congruence,

$△DEF\cong △AFE$

Similarly, $△DEF\cong △FBD$ and $△DEF\cong △EDC$

Hence, $△AFE\cong △FBD\cong △EDC\cong △DEF$