Question

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

Answer 1

We have, D and E as the mid-points of AB and AC

So, according to the mid-point theorem

DE || BC and DE `=1/2`BC            ...(i)

In ΔADE and ΔABC

∠A = ∠A [Common]

∠ADE = ∠B [Corresponding angles]

Then, ΔADE ~ ΔABC [By AA similarity]

By area of similar triangle theorem

`("Area"(triangleADE))/("Area"(triangleABC))="DE"^2/"BC"^2`

`=(1/2BC)^2/"BC"^2`               [From (i)]

`=(1/4BC^2)/"BC"^2`

`=1/4`