Question

As medidas dos lados de um triângulo são indicadas, em centímetro, pelas expressões 4x -1, 3x + 3 e x² + 4 e formam uma PA, nessa ordem. Calcule o perímetro desse triângulo

a)26cm
b)36cm
c)42cm
d)52cm

Answer 1

$\Large\boxed{\begin{array}{l}\sf PA\,(4x-1,3x+3,x^2+4)\\\sf r=3x+3-(4x-1)\\\sf r=3x+3-4x+1\\\sf r=-x+4\\\sf r=x^2+4-(3x+3)\\\sf r=x^2+4-3x-3\\\sf r=x^2-3x+1\end{array}}$$\Large\boxed{\begin{array}{l}\sf Como\,r=r\,podemos\,escrever:\\\sf x^2-3x+1=-x+4\\\sf x^2-3x+x+1-4=0\\\sf x^2-2x-3=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-2)^2-4\cdot1\cdot(-3)\\\sf\Delta=4+12\\\sf\Delta=16\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-2)\pm\sqrt{16}}{2\cdot1}\\\\\sf x=\dfrac{2\pm4}{2}\begin{cases}\sf x_1=\dfrac{2+4}{2}=\dfrac{6}{2}=3\\\\\sf x_2=\dfrac{2-4}{2}=-\dfrac{2}{2}=-1\end{cases}\\\\\sf como\,x > 0,ent\tilde ao\,x=3\end{array}}$

$\Large\boxed{\begin{array}{l}\sf 4x-1=4\cdot3-1=12-1=11\,cm\\\sf 3x+3=3\cdot3+3=9+3=12\\\sf x^2+4=3^2+4=9+4=13\\\sf O\,per\acute imetro\,\acute e\,a\,soma\,de\,todos\,os\,lados.\\\sf p=11+12+13\\\sf p=36\,cm\\\\\huge\boxed{\boxed{\boxed{\boxed{\rm\maltese\,alternativa\,b}}}}\end{array}}$