Question

A medida dos lados de uma triângulo são (2x + 2), (4x) e (2x ao quadrado - 10) formando nessa ordem uma PA. Determine o perímetro desse triângulo em centímetros

Answer 1

$\boxed{\begin{array}{l}\sf(2x-10)^2+2x+2=2\cdot4x\\\sf 4x^2-40x+100+</p><p>2x+2-8x=0\\\sf 4x^2-46x+102=0\div2\\\sf 2x^2-23x+51=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-23)^2-4\cdot2\cdot51\\\sf\Delta=529-408\\\sf\Delta=121\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-23)\pm\sqrt{121}}{2\cdot2}\\\\\sf x=\dfrac{23\pm11}{4}\begin{cases}\sf x_1=\dfrac{23+11}{4}=\dfrac{34}{4}=8,5\\\\\sf x_2=\dfrac{23-11}{4}=\dfrac{12}{4}=3\end{cases}\end{array}}$

$\boxed{\begin{array}{l}\boxed{\sf para ~x=8,5:}\\\sf2\cdot8,5+2=19~cm\\\sf(2\cdot8,5-10)^2=49~cm\\\sf4\cdot8,5=34~cm\\\sf 2p\longrightarrow per\acute imetro\\\sf 2p=19+49+34=102~cm\\\boxed{\sf para~x=3:}\\\sf 2\cdot3+2=8~cm\\\sf (2\cdot3-10)^2=16~cm\\\sf 4\cdot3=12~cm\\\sf 2p=8+16+12=36~cm\end{array}}$