Question

In the adjoining figure, BD || CA, E is the midpoint of CA and BD = ½ CA. Prove that ar (△ ABC) = 2 ar (△ DBC).

Answer 1

It is given that E is the midpoint of CA and BD = ½ CA

So we get

BD = CE

We know that BD || CA

BD = CE

In the same way BD || CE

BD = CE

Hence, BCED is a parallelogram.

We know that △ DBC and △ EBC lie on the same base and between the same parallel lines

So we get

Area of △ DBC = Area of △ EBC ….. (1)

From the figure we know that BE is the median of △ ABC

We get

Area of △ BEC = ½ (Area of △ ABC)

Using equation (1) we get

Area of △ DBC = ½ (Area of △ ABC)

So we get

Area of △ ABC = 2 (Area of △ DBC)

Therefore, it is proved that ar (△ ABC) = 2 ar (△ DBC).