Question

Determine o valor de x do triângulo abaixo.

Answer 1

1º vamos achar o último ângulo que falta :

$\displaystyle \sf 105^\circ+45^\circ+\theta =180^\circ \\\\ 150^\circ +\theta=180^\circ \\\\ \theta = 180^\circ-150^\circ \\\\ \theta = 30^\circ$

2º Aplicando Lei dos senos :

$\displaystyle \sf \frac{x}{sen(\theta)} = \frac{6}{sen(45^\circ)} \\\\\\ \frac{x}{sen(30^\circ)}=\frac{6}{sen(45^\circ)} \\\\\\ x = \frac{sen(30^\circ)\cdot 6}{sen(45^\circ)}\\\\\\ x = \frac{\displaystyle \frac{1}{2}\cdot 6 }{\displaystyle \frac{\sqrt{2}}{2}} \to x = \frac{6}{\sqrt{2}} \to x=\frac{6}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\\\\\ x = \frac{6\sqrt{2}}{2} \\\\\\\ \huge\boxed{\sf x=3\sqrt{2}\ }\checkmark$