Q16 of 77 Page 15

ABCD is a parallelogram whose diagonals AC and BD intersect at O. A line through O intersects AB at P and DC at Q. Prove that

ar(Δ POA) = ar(Δ QOC)

In ΔPOA and QOC, we have

∠AOP = ∠COQ (Vertically opposite angle)


OA = OC (Diagonals of parallelogram bisect each other)


∠PAC = ∠QCA (AB ‖ DC, alternate angles)


So, by ASA congruence rule, we have


ΔPOA ≅ ΔQOC


Area (ΔPOA) = Area (ΔQOC)


Hence, proved



More from this chapter

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14

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)

 15

ABCD is a parallelogram in which BC is produced to E such that CE=BC. AE intersects CD at F.

(i) Prove that ar(Δ ADF) = ar(Δ ECF)


(ii) If the area of Δ DFB=3 cm2, find the area of ||gm ABCD.

 17

ABCD is a parallelogram. E is a point on BA such that BE = 2 EA and F is a point on DC, such that DF=2FC. Prove that AE CF is a parallelogram whose area is one third of the area of parallelogram ABCD.

 18

In a Δ ABC, P and Q are respectively the mid-point of AB and BC and R is the mid-point of AP. Prove that:

(i) ar(Δ PBQ) = ar(Δ ARC)


(ii) ar(Δ PQR) = ar(Δ ARC)


(iii) ar(Δ RQC) = ar(Δ ABC)