Question

D is the midpoint of side BC of △ ABC and E is the midpoint of BD. If O is the midpoint of AE, prove that ar (△ BOE) = 1/8 ar (△ ABC).

Answer 1

From the figure we know that O is the midpoint of AE.

We know that BO is the median of △ BAE

So we get

ar (△ BOE) = ½ ar (△ ABE) ……… (1)

We know that E is the midpoint of BD

AE divides the △ ABD into two triangles of equal area

So we get

ar (△ ABE) = ½ ar (△ ABD) ……… (2)

We know that D is the midpoint of BC

So we get

ar (△ ABD) = ½ ar (△ ABC) ……… (3)

We know that

ar (△ BOE) = ½ ar (△ ABE) using equation (1)

Substituting equation (2) in (1)

ar (△ BOE) = ½ (½ ar (△ ABD))

So we get

ar (△ BOE) = ¼ ar (△ ABD)

By substituting equation (3)

ar (△ BOE) = ¼ (½ ar (△ ABC))

So we get

ar (△ BOE) = 1/8 ar (△ ABC)

Therefore, it is proved that ar (△ BOE) = 1/8 ar (△ ABC).