Question

ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove that:

(i) ar (Δ ADO) = ar(Δ CDO)

(ii) ar (Δ ABP) = ar (Δ CBP).

Answer 1

Given: Here from the given figure we get

(1) ABCD is a parallelogram

(2) BD and CA are the diagonals intersecting at O.

(3) P is any point on BO

To prove:

(a) Area of ΔADO = Area ofΔ CDO

(b) Area of ΔAPB = Area ofΔ CBP

Proof: We know that diagonals of a parallelogram bisect each other.

O is the midpoint of AC and BD.

Since medians divide the triangle into two equal areas

In ΔACD, DO is the median

Area of ΔADO = Area ofΔ CDO

Again O is the midpoint of AC.

In ΔAPC, OP is the median

⇒ Area of ΔAOP = Area of ΔCOP …… (1)

Similarly O is the midpoint of AC.

In ΔABC, OB is the median

⇒ Area of ΔAOB = Area of ΔCOB …… (2)

Subtracting (1) from (2) we get,

Area of ΔAOB − Area of ΔAOP = Area of ΔCOB − Area of ΔCOP

⇒ Area of ΔABP = Area of ΔCBP

Hence it is proved that

(a)

(b)