Question

In ΔABC, D and E are the midpoints of AB and AC respectively. Find the ratio of the areas of ΔADE and ΔABC. 

 

Answer 1

It is given that D and E are midpoints of AB and AC.
Applying midpoint theorem, we can conclude that DE ‖ BC.
Hence, by B.P.T., we get : 

`(AD)/(AB)=(AE)/(AC)` 

Applying SAS similarity theorem, we can conclude that Δ ADE~ Δ ABC.
Therefore, the ration of areas of these triangles will be equal to the ratio of squares of their corresponding sides. 

∴ `(9ar(ΔADE))/(ar(ΔABC))=(DE)^2/(BC)^2` 

= `(1/2 BC)^2/(BC)^2` 

= `1/4`