Question

ABC is a triangle in which D is the midpoint of BC and E is the midpoint of AD.
Prove that $\mathrm{ar}(∆BED)=\frac{1}{4}\mathrm{ar}(∆ABC)$.

Answer 1

Given: D is the midpoint of BC and E is the midpoint of AD.
To prove: ar(∆BED) =$\frac{1}{4}$ ar(∆ABC)
Proof:
Since D is the mid point of BC, AD is 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 areas.
i.e., ar(∆ABD) = $\frac{1}{2}$ar(∆ABC) ...(i)

Also, 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)