Respostas

2013-08-07T22:36:07-03:00
Primeiramente, utilizaremos a Lei dos Cossenos para achar o valor de x:


a^2=b^2+c^2-2\cdot b\cdot c\cdot\cos(\alpha)\\\\
(x+2)^2=x^2+(x+1)^2-2\cdot x\cdot(x+1)\cdot\cos(120^{\circ})\\\\
x^2+4x+4=x^2+x^2+2x+1-2\cdot (x^2+x)\cdot(-\dfrac{1}{2})\\\\
x^2+4x+4=x^2+x^2+2x+1+x^2+x\\\\
2x^2-x-3=0\\\\\\
\Delta=b^2-4\cdot a\cdot c\\
\Delta=(-1)^2-4\cdot2\cdot(-3)\\
\Delta=1+24\\
\Delta=25\\\\
x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\Longrightarrow x=\dfrac{1\pm\sqrt{25}}{2\cdot2}\Longrightarrow x=\dfrac{1\pm5}{4}

Como x tem de ser positivo:

x=\dfrac{1+5}{4}=\dfrac{6}{4}=\dfrac{3}{2}

Calculando o perímetro, temos:

2p=x+x+2+x+1\\\\
2p=\dfrac{3}{2}+\dfrac{3}{2}+2+\dfrac{3}{2}+1\\\\
2p=1,5+1,5+2+1,5+1\\\\
2p=7,5
4 3 4