D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
Solution.
Given: In , D, E and F are respectively the mid-points of the sides AB, BC and CA.
To prove: is divided into four congruent triangles.
Proof: Using given conditions we have
Using mid-point theorem
and
and
and
In and
AD = EF
AF = DE
DF = FD {common side}
{by SSS congruence}
Similarly we can prove that,
So, is divided into four congruent triangles
Hence proved