In triangle ABC, D, E and F are respectively the mid-points of sides AB, BC, CA show that triangle - Maths - Quadrilaterals

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

Poonam Mittal · Accepted Answer

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

$i.e. Δ BDE ≅ Δ DEF ≅ Δ CEF ≅ Δ ADF$

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

⇒ΔBDE $≅$ ΔDEF (Since diagonal of a parallelogram divides it into two congruent triangles) ......(3)

Similarly

ΔDEF $≅$ ΔCEF ......(4)

ΔDEF $≅$ ΔADF ......(5)

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

We have,

$Δ BDE ≅ Δ DEF ≅ Δ CEF ≅ Δ ADF$

Hence proved

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.