Question

In ​∆ABC, if E is the mid-point of median AD, prove that $\mathrm{ar}(∆BED)=\frac{1}{4}\mathrm{ar}(∆ABC)$.

Answer 1

Given: D is the mid-point of BC and E is the mid-point of AD.
To prove: ar(∆BED) =$\frac{1}{4}$ ar(∆ABC)

Proof:
Since, D is the mid-point of BC, AD is the median of ∆ ABC and BE is the median of ∆ ABD.
We know that a median of a triangle divides it into two triangles of equal area.
So, ar (∆ ABD ) = $\frac{1}{2}$ ar (∆ ABC).................(i)

So, ar (∆ BED ) =$\frac{1}{2}$ar (∆ ABD).................(ii)

From (i) and (ii), we have:
ar (∆ BED ) = $\frac{1}{2}$​​ ⨯ $\frac{1}{2}$⨯​ ar (∆ ABC)
∴​ ar (∆ BED )​ =$\frac{1}{4}$ ⨯ ar(∆ABC)