Question

D, E and F are respectively the mid-points of the sides AB, BC and CA of a ΔABC. Prove that by joining these mid-points D, E and F, the ΔABC is divided into four congruent triangles.

Answer 1

ABC is a triangle and D, E and F are mid-points of sides AB, BC and CA, respectively. Then,

AD = BD = 1/2AB

BE = EC = 1/2BC

And AF = CF = 1/2AC

Now, by mid-point theorem,

EF||AB and EF = 1/2AB = AD = BD

ED||AC and ED = 1/2AC = AF = CF

DF||BC and DF = 1/2BC = BE = CE

Now, in ΔADF and ΔEFD,

AD = EF

AF = DE

And DF = FD (common)

ΔADF ≅ ΔEFD

Similarly, ΔDEF ≅ ΔDEB

And ΔDEF ≅ ΔCEF

Thus, ΔABC is divided into four congruent triangles.