Question

D and E are the mid point of the sides AB and AC of a triangle ABC. Then find the ratio of the areas of the triangle ADE and ABC.

Answer 1

APPLYING MIDPOINT THEOREM
According to the Midpoint Theorem:
“The line segment joining the midpoints of two sides of any triangle, is parallel to the third side, and also half of that third side.”
By applying this theorem, we can say that:

ED =BC/2

Now the other sides
AE=AC/2
AD=AB/2
(Since D and E are midpoints)

Here by the property of triangle we can say that the triangles are similiar.
Since their sides are in ratio.
It is known that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
ie
(Area of ABC / AED) = BC²/ED²
= (2ED)²/(ED)²
since BC =2ED
Area ratio = 4

Area of ∆ABC/∆ADE = 4