Question

Qual deve ser o valor de x no triângulo abaixo? *

a) 0.5
b) 1,5
c) 2,5O
d) 3,5
e) 4,5​

Answer 1

Lei dos Cossenos

${a}^{2} = {b}^{2} + {c}^{2} - 2.b.c.\cos \alpha$

Daí,

${(x+2)}^{2} = {x}^{2} + {(x+1)}^{2} -2 x(x+1). \cos \alpha \\ \\ {x}^{2} + 4x + 4 = {x}^{2} + {x}^{2} + 2x + 1 - 2x( x + 1)cos \alpha \\\\ cos \alpha = \frac{ {x}^{2} + 4x + 4 -2{x}^{2} - 2x - 1}{ -2 x(x+ 1)} \\ \\ cos \alpha = \frac{ -{x}^{2} + 2x + 3}{ -2x(x + 1)} \\ \\ cos \alpha = \frac{ - (x - 3)(x + 1)}{-2x(x + 1)} = \frac{(x - 3)}{2x}$

Assim,

$cos \: 120° = \frac{(x - 3)}{2x} \\ \\ - \frac{1}{2} = \frac{(x - 3)}{2x} \\ \\ -2x = 2x - 6 \\\\4x=6\\\\x=\frac{6}{4}= 1,5$