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Question

If the medians of ΔABC intersects at G, then ar(ΔAGB)=ar(ΔAGC)=ar(ΔBGC)=1kar(ΔABC).Find k

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Solution

Given:ThemediansAD,BEandCFofABCintersectsatGWeknowthatmediansofatriangledividesitintotwotrianglesofequalarea.InABC,ADisthemedianar(ABD)=ar(ACD).......(1)InBGC,GDisthemedianar(GBD)=ar(GCD).....(2)Substracting(2)from(1),wegetar(ABD)ar(GBD)=ar(ACD)ar(GCD)ar(AGB)=ar(AGC)......(3)Similarly,ar(AGB)=ar(GBC).....(4)From(3)and(4)wegetar(AGB)=ar(AGC)=ar(BGC)But,ar(AGB)+ar(AGC)+ar(BGC)=ar(ABC)3ar(AGB)=ar(ABC)ar(AGB)=13ar(ABC)Hence,ar(AGB)=ar(AGC)=ar(BCG)=13ar(ABC)
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