Question

In triangle ABC, D, E and F are respectively the mid-points of sides AB, BC, CA. show that triangle ABC is divided into four congruent triangles by joining D, E, and F.

Answer 1

Given that D, E and F are the mid points of sides AB, BC, CA respectively.

To show: ΔABC is divided into four congruent triangles



Proof: D is the mid point of AB

F is the mid point of AC.

∴DF||BC (By mid point theorem)

⇒DF||BE ......(1)

also E is the mid point of BC

and F is the mid point of AC.

∴EF||AB (By mid point theorem)

⇒EF||DB ......(2)

By (1) & (2)

BEFD is a parallelogram

⇒ΔBDEΔDEF (Since diagonal of a parallelogram divides it into two congruent triangles) ......(3)

Similarly

ΔDEFΔCEF ......(4)

ΔDEFΔADF ......(5)

By (3), (4) & (5)

We have,



Hence provedGiven that D, E and F are the mid points of sides AB, BC, CA respectively.

To show: ΔABC is divided into four congruent triangles



Proof: D is the mid point of AB

F is the mid point of AC.

∴DF||BC (By mid point theorem)

⇒DF||BE ......(1)

also E is the mid point of BC

and F is the mid point of AC.

∴EF||AB (By mid point theorem)

⇒EF||DB ......(2)

By (1) & (2)

BEFD is a parallelogram

⇒ΔBDEΔDEF (Since diagonal of a parallelogram divides it into two congruent triangles) ......(3)

Similarly


ΔDEFΔCEF ......(4)

ΔDEFΔADF ......(5)

By (3), (4) & (5)

We have,



Hence proved