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Question Number 106123 by DeepakMahato last updated on 03/Aug/20

$$Ina△ABCwithsides\mathit{a}=6cm,\mathit{b}=?and\mathit{c}=?$$$$areais270{cm}^2.Andif$$$$midlineofthe△is7cm$$$$findthevalueofsides\mathit{b}\mathit{a}\mathit{n}\mathit{d}\mathit{c}.$$$$$$$$\mathit{P}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}\mathit{s}\mathit{o}\mathit{l}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{i}\mathit{s}\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathit{a}\mathit{l}\mathit{i}\mathit{t}\mathit{t}\mathit{l}\mathit{e}$$$$\mathit{e}\mathit{x}\mathit{p}\mathit{l}\mathit{a}\mathit{n}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}.$$

Commented by PRITHWISH SEN 2 last updated on 03/Aug/20

$$\because \mathrm{a}=6\mathrm{cm}\mathrm{and}\mathrm{the}\mathrm{midline}\mathrm{is}7\mathrm{cm}$$$$\therefore \mathrm{the}\mathrm{midline}\mathrm{must}\mathrm{be}\mathrm{parallal}\mathrm{to}\mathrm{b}\mathrm{or}\mathrm{c}$$$$\therefore \mathrm{b}\mathrm{or}\mathrm{c}=14\mathrm{cm}\mathrm{and}\mathrm{you}\mathrm{may}\mathrm{consider}\mathrm{any}\mathrm{of}\mathrm{the}$$$$\mathrm{b}\mathrm{or}\mathrm{c}\mathrm{side}\mathrm{It}{\mathrm{does}}^{\prime }\mathrm{nt}\mathrm{affect}\mathrm{the}\mathrm{overall}\mathrm{calculation}.$$

Commented by 1549442205PVT last updated on 03/Aug/20

$$\mathrm{Question}\mathrm{lead}\mathrm{to}\mathrm{a}\mathrm{cotradiction}\mathrm{thing}$$$$\mathrm{Indeed},\mathrm{the}\mathrm{midpoint}\mathrm{l}\mathrm{must}\mathrm{be}//\mathrm{b}\mathrm{or}$$$$//\mathrm{c}\mathrm{and}\mathrm{then}\mathrm{b}=2\times 7=14\mathrm{cm}\mathrm{or}$$$$\mathrm{c}=2\times 7\mathrm{cm}=14\mathrm{cm}$$$$\mathrm{i})\mathrm{If}\mathrm{l}//\mathrm{b}\mathrm{then}\mathrm{b}=14\mathrm{cm}\mathrm{and}{\mathrm{S}}_{\mathrm{\Delta }\mathrm{ABC}}=\frac{1}{2}\mathrm{absinC}$$$$=\frac{7\times 6\times \mathrm{sinC}}{2}=21\mathrm{sinC}⩽21<<270\left({\mathrm{cm}}^2\right)$$$$\mathrm{ii})\mathrm{If}\mathrm{l}//\mathrm{c}\mathrm{thi}\mathrm{c}=2\mathrm{l}=14\left(\mathrm{cm}\right)$$$$\Rightarrow {\mathrm{S}}_{\mathrm{\Delta }\mathrm{ABC}}=\frac{\mathrm{acsinB}}{2}=21\mathrm{sinB}⩽21<<270$$$$\mathrm{In}\mathrm{other}\mathrm{words},\mathrm{there}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{exist}$$$$\mathrm{any}\mathrm{triangle}\mathrm{that}\mathrm{satisfying}\mathrm{the}$$$$\mathrm{conditions}\mathrm{of}\mathrm{given}\mathrm{problem}!$$

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