Given
D,E and F are respectively the midpoints of sides AB,BC and CA of ΔABC
To find
ar(△DEF)ar(△ABC)
We know that
The line segment joining the midpoints of any two sides of a triangle is half the third side and parallel to it.
∴FD=12BC,ED=12ACandEF=12AB
In △ABC and △EFD, we have
ABEF=BCFD=ACED=2...(i)
⇒△ABC∼△EFD[by SSS similarity criterion]
Also, We know that
If two triangles are similar, then the ratio of the area of both triangles is equal to the square of the ratio of their corresponding sides
∴ar(△ABC)ar(△EFD)=(ABEF)2=4[from (i)]
⇒ar(△EFD)ar(△ABC)=14
Hence, the ratio of the areas of △DEF and △ABC is 1:4