Question

in triangle abc,def are respectively the midpoints of sides ab,bc,ca show that triangle abc is divided into four congruent triangle by joining d,e,f​

Answer 1

Answer:

Given:

A ΔABC and ΔDEF which is formed by joining the mid-point D,E and F of the sides AB,BC and CA of ΔABC

Refer image,

To prove: ΔDEF≅ΔEDB≅ΔCFE≅ΔFAD

Proof: DF joins mid-points of sides AB and AC respectively of ΔABC

∴DF∣∣BC [Mid-point theorem]

⇒DF∣∣BE

SImilarly, EF∣∣BD

So, quadrilateral BEFD is a parallelogram.

⇒ΔDEF≅ΔEDB.........(1)

(∵ DIagonal of parallelogram divides it into two congruent triangles)

Similarly, ΔDEF≅ΔCFE......(2)

and ΔDEF≅ΔFAD..........(3)

From equation (1),(2) and (3), we get

ΔDEF≅ΔEDB≅ΔCFE≅ΔFAD