Question

In the adjoining figure, ABCD is a parallelogram and O is any point on the diagonal AC. Show that ar(∆AOB) = ar(∆AOD).

Answer 1

Given: ABCD is a parallelogram and O is any point on AC.
To prove: ar(∆AOB) = ar(∆AOD)
Proof:
Since the diagonals of a parallelogram bisect each other, P is the midpoint of BD.
Now, AP is a median of ∆BAD.
i.e., ar(∆PAB ) = ar(∆PAD) ...(i)
OP is median of ∆ODB
i.e., ar(∆OPB ) = ar(∆ODP) ...(ii)

Adding (i) and (ii), we have:
ar(∆PAB) + ar(∆OPB) = ​ar(∆PAD)​ + ar(∆ODP)
∴​​ ar(∆AOB)​ = ar(∆AOD)