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

In a △ABC with sides a=6cm,b=? and c=? area is 270cm^2 .And if midline of the △ is 7 cm find the value of sides b and c. Please solve this with a little explanation.

$${In}\:{a}\:\bigtriangleup{ABC}\:{with}\:{sides}\:\boldsymbol{{a}}=\mathrm{6}{cm},\boldsymbol{{b}}=?\:{and}\:\boldsymbol{{c}}=? \\ $$$${area}\:{is}\:\mathrm{270}{cm}^{\mathrm{2}} .{And}\:{if}\: \\ $$$${midline}\:{of}\:{the}\:\bigtriangleup\:{is}\:\mathrm{7}\:{cm} \\ $$$${find}\:{the}\:{value}\:{of}\:{sides}\:\:\boldsymbol{{b}}\:\boldsymbol{{and}}\:\boldsymbol{{c}}. \\ $$$$ \\ $$$$\boldsymbol{{Please}}\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\:\boldsymbol{{with}}\:\boldsymbol{{a}}\:\boldsymbol{{little}} \\ $$$$\boldsymbol{{explanation}}. \\ $$

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

∵ a=6cm and the midline is 7 cm ∴ the midline must be parallal to b or c ∴ b or c = 14 cm and you may consider any of the b or c side It does′nt affect the overall calculation.

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

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

Question lead to a cotradiction thing Indeed,the midpoint l must be //b or //c and then b=2×7=14 cm or c=2×7 cm =14 cm i)If l//b then b=14 cm and S_(ΔABC) =(1/2)absinC =((7×6×sinC)/2)=21sinC≤21<<270 (cm^2 ) ii)If l//c thi c=2l=14(cm) ⇒S_(ΔABC) =((acsinB)/2)=21sinB≤21<<270 In other words,there isn′t exist any triangle that satisfying the conditions of given problem!

$$\mathrm{Question}\:\mathrm{lead}\:\mathrm{to}\:\mathrm{a}\:\mathrm{cotradiction}\:\mathrm{thing} \\ $$$$\mathrm{Indeed},\mathrm{the}\:\mathrm{midpoint}\:\mathrm{l}\:\:\mathrm{must}\:\mathrm{be}\://\mathrm{b}\:\mathrm{or} \\ $$$$//\mathrm{c}\:\mathrm{and}\:\mathrm{then}\:\mathrm{b}=\mathrm{2}×\mathrm{7}=\mathrm{14}\:\mathrm{cm}\:\mathrm{or} \\ $$$$\mathrm{c}=\mathrm{2}×\mathrm{7}\:\mathrm{cm}\:=\mathrm{14}\:\mathrm{cm} \\ $$$$\left.\mathrm{i}\right)\mathrm{If}\:\mathrm{l}//\mathrm{b}\:\mathrm{then}\:\mathrm{b}=\mathrm{14}\:\mathrm{cm}\:\mathrm{and}\:\mathrm{S}_{\Delta\mathrm{ABC}} =\frac{\mathrm{1}}{\mathrm{2}}\mathrm{absinC} \\ $$$$=\frac{\mathrm{7}×\mathrm{6}×\mathrm{sinC}}{\mathrm{2}}=\mathrm{21sinC}\leqslant\mathrm{21}<<\mathrm{270}\:\left(\mathrm{cm}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{ii}\right)\mathrm{If}\:\mathrm{l}//\mathrm{c}\:\mathrm{thi}\:\mathrm{c}=\mathrm{2l}=\mathrm{14}\left(\mathrm{cm}\right) \\ $$$$\Rightarrow\mathrm{S}_{\Delta\mathrm{ABC}} =\frac{\mathrm{acsinB}}{\mathrm{2}}=\mathrm{21sinB}\leqslant\mathrm{21}<<\mathrm{270} \\ $$$$\mathrm{In}\:\:\mathrm{other}\:\mathrm{words},\mathrm{there}\:\mathrm{isn}'\mathrm{t}\:\mathrm{exist}\: \\ $$$$\mathrm{any}\:\mathrm{triangle}\:\mathrm{that}\:\mathrm{satisfying}\:\mathrm{the}\: \\ $$$$\mathrm{conditions}\:\mathrm{of}\:\mathrm{given}\:\mathrm{problem}! \\ $$

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