Question

please draw the diagram

IF D, E, F are the mid-points of sides BC, CA and AB respectively of triangle ABC, then the ratio of the areas of triangles DEF and ABC is.​

Answer 1

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≅Δ EDB (1)

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

Similarly, ΔDEF≅Δ CFE (2)

and ΔDEF≅Δ FAD (3)

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

ΔDEF≅ΔEDB≅ΔCFE≅ΔFAD

Hence, proved.