Solution
Maan lijiye △PQR ka kul area S hai. Hume △ABC ka area nikalne ke liye pure triangle mein se kone wale teen chhote triangles (△PQC,△QAB,△RCA) ka area ghatana hoga.
1. Side ka ratio nikalna:
Diya gaya hai ki ARQA=21, iska matlab hai ki QA=31QR aur AR=32QR.
Isi tarah:
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RB=31RP aur BP=32RP
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PC=31PQ aur CQ=32PQ
2. Chhote triangles ka area:
Kisi bhi triangle ka area do sides aur unke beech ke angle se nikala ja sakta hai: Area=21absin(θ).
△PQR ke muqable chhote triangles ka area:
Ek triangle ka area nikalte hain, jaise △PCB:
Area(△PCB)=PQPC×PRPB×Area(△PQR)
Area(△PCB)=31×32×S=92S
Chunki symmetrical ratio hai, isliye:
Area(△QAC)=QRQA×QPQC×S=31×32×S=92S
Area(△RBA)=RPRB×RQRA×S=31×32×S=92S
3. △ABC ka area:
Area(△ABC)=Area(△PQR)−[Area(△PCB)+Area(△QAC)+Area(△RBA)]
Area(△ABC)=S−(92S+92S+92S)
Area(△ABC)=S−96S=S−32S=31S
4. Final Ratio:
Area(△ABC)Area(△PQR)=31SS=3
Sahi Option: (B) 3
