Question

Helpp, resposta gabarito: $x= 3( \sqrt{2}+ \sqrt{6})cm; y=6 \sqrt{2} cm$

Answer 1

Lei dos senos:

$\frac{y}{sen\ 45^{\circ}} = \frac{6}{sen \ 30^{\circ}}$

$\frac{y}{ \frac{ \sqrt{2} }{2} } = \frac{6}{ \frac{1}{2} } \\\\ \frac{y}{ \frac{ \sqrt{2} }{2} } = \frac{6}{ \frac{1}{2} } \\\\ \frac{y}{ \frac{ \sqrt{2} }{2} } = 12\\\\ y=12* \frac{\sqrt{2}}{2} \\\\ \boxed{y=6\sqrt{2}\ cm}$

O ângulo X vale :

X + 30° + 45° = 180°
X = 180° - 75°
X = 105°

sen α = sen (180-α)
sen 75° = sen (180°-75°)
sen 75° = sen (105°)

$\frac{x}{sen\ 75^{\circ}} = \frac{6}{ sen\ 30^{\circ} } \\\\ \frac{x}{ \frac{ \sqrt{2}+ \sqrt{6} }{4} } = \frac{6}{ \frac{1}{2} } \\\\ \frac{x}{ \frac{ \sqrt{2}+ \sqrt{6} }{4} } = 12 } \\\\ x=(\frac{ \sqrt{2}+ \sqrt{6} }{4} ) *12\\\\ \boxed{x=3(\sqrt{2}+ \sqrt{6})\ cm }$