Question

In the adjoining figure, △ ABC and △ DBC are on the same base BC with A and D on opposite sides of BC such that ar (△ ABC) = ar (△ DBC). Show that BC bisects AD.

Answer 1

Construct AP ⊥ BC and DQ ⊥ BC

We know that

Area of △ ABC = ½ × BC × AP

In the same way

Area of △ BCD = ½ × BC × DQ

Equating both we get

½ × BC × AP = ½ × BC × DQ

So we get

AP = DQ ……. (1)

Consider △ AOP and △ QOD

From the figure we know that

∠ APO = ∠ DQO = 90^o

We know that ∠ AOP and ∠ DOQ are vertically opposite angles

∠ AOP = ∠ DOQ

By AAS congruence criterion

△ AOP ≅ △ QOD

OA = OD (c. p. c. t)

Therefore, it is proved that BC bisects AD.