If D, E, F are the respectively the midpoints of sides BC, CA and AB of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.

 

 

 

 

 

Answer:

Using midpoint theorem,

The segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

∴ DF || BC

The opposite sides of the quadrilateral are parallel and equal.

BDFE is a parallelogram

DFCE is a parallelogram.

In ∆ABC and ∆EFD,

∠???????????? = ∠???????????? (???????????????????????????????? ???????????????????????? ???????? ???? ????????????????????????????????????????????????????)

∠???????????? = ∠???????????? (???????????????????????????????? ???????????????????????? ???????? ???? ????????????????????????????????????????????????????)

By AA similarity criterion,

∆ABC ~ ∆EFD

If two triangles are similar, then the ratio of their areas is equal to the squares of their corresponding sides.

Hence, the ratio of the areas of ∆DEF and ∆ABC is 1 : 4.