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Question

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB//DC, show that :
Area(ΔDCB)=Area(ΔACB).

196253.png

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Solution

Let us draw DNAC and BMAC as shown in above figure.

In triangles DON and BMO,

DNO=BMO ( by construction)

DON=BOM ( vertically opposite angles)

OD=OB (given)

By AAS congruence rule,

ΔDONΔBOM

DN=BM.............( eqn 1)

We know that congruent triangles have equal areas, thus, Ar(ΔDON)=Ar(ΔBOM).

In triangles DNC and BMA,

DNC=BMA ( by construction)

CD=AB (given)

DN=BM ( using eqn 1)

ΔDNCΔBMA ( RHS congruence rule)

Therefore, Ar(ΔDOC)=Ar(ΔAOB).

Adding areas on both sides, we get

Ar(ΔDNC)+Ar(ΔOCB)=Ar(ΔAOB)+Ar(ΔOCB)

Ar(ΔDCB)=Ar(ΔACB)

Hence proved.

407365_196253_ans.png

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