Construct AE ⊥ BC
We know that
Area of △ ABD = ½ × BD × AE ……. (1)
Area of △ ABC = ½ × BC × AE …….. (2)
It is given that
BD = ½ BC
We get
BC = BC + DC
It can be written as
BC = BD + 2BD
So we get
BC = 3BD
By division
BD = 1/3 BC …….. (3)
Using equation (1) we get
Area of △ ABD = ½ × BD × AE
By substituting (3)
Area of △ ABD = ½ × 1/3 × BC × AE
So we get
Area of △ ABD = 1/3 (1/2 × BC × AE)
Substituting equation (2)
Area of △ ABD = 1/3 × Area of △ ABC
Therefore, it is proved that ar (△ ABD) = 1/3 × ar (△ ABC).