Question

Fie a,b,c lungimile laturilor unui triunghi, Aflati perimetrul si aria triunghiului stiind ca a,b,c satisface relatia a^2+b^2+c^2+110=2(5a+6b+7c)

Answer 1

 

$\displaystyle\bf\\a^2+b^2+c^2+110=2(5a+6b+7c)\\\\a^2+b^2+c^2+110-2(5a+6b+7c)=0\\\\a^2+b^2+c^2-2(5a+6b+7c)+110=0\\\\110=25+36+49\\\\a^2+b^2+c^2-2(5a+6b+7c)+25+36+49=0\\\\a^2+b^2+c^2-2\times5a+2\times6b+2\times7c+25+36+49=0\\\\(a^2-2\times5a+25)+(b^2-2\times6b+36)+(c^2-2\times7c+49)=0\\\\(a-5)^2+(b-6)^2+(c-7)^2=0\\\\\implies~(a-5)^2=0~~si~~(b-6)^2=0~~si~~(c-7)^2=0\\\\\implies~(a-5)=0~~si~~(b-6)=0~~si~~(c-7)=0$

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$\displaystyle\bf\\\text{Propun sa folosim centimetrul ca unitate de masura.}\\a=5~cm\\b=6~cm\\c=7~cm\\\\\text{Calculam perimetrul triunghiului:}\\\\P=a+b+c=5+6+7=18~cm\\\boxed{\bf~P=18~cm}\\\\\text{Calculam aria triunghiului cu formula lui Heron.}\\P=18~cm\\\\p=\frac{P}{2}=\frac{18}{2}=9~cm~~(semiperimetrul.)\\\\A=\sqrt{p(p-a)(p-b)(p-c)}\\\\A=\sqrt{9(9-5)(9-6)(9-7)}\\\\A=\sqrt{9\times4\times3\times2}\\\\A=\sqrt{36\times6}\\\\\boxed{\bf~A=6\sqrt{6}~cm^2}$