Answer both with STEPS

Answer:7 )x =  [tex]\frac{3\sqrt{2} }{2}[/tex][tex]y= 3[/tex]8 )[tex]x=6\sqrt{6}[/tex][tex]y= 9\sqrt{2}[/tex]Step-by-step explanation:7  )                                 8)In Δ ABC                                      In Δ XYZ             ∠ C = 45°                                          ∠ X = 60°∠ A = 90°                                           ∠ Y = 90°[tex]AC= \frac{3\sqrt{2} }{2}[/tex]             [tex]XY= 3\sqrt{6}[/tex]To Find :x = ?y = ?Solution:We Know In Δ ABC∠ C = 45°∠ A = 90°∴ ∠ B = 45°  ......Angle sum property of a triangle i.e 180°∴  Δ ABC is an Isosceles Triangle∴ AC = AB = x =  [tex]\frac{3\sqrt{2} }{2}[/tex]Now appplying Trignometry identity we get[tex]\sin C = \frac{\textrm{side opposite to angle C}}{Hypotenuse}\\\\\sin 45 = \frac{AC}{BC}\\\\\frac{1}{\sqrt{2} } =\frac{\frac{3\sqrt{2} }{2}}{y}\\\\y=\frac{3\times \sqrt{2}\times \sqrt{2}  }{2}\\\\y= 3[/tex]Now In Δ XYZ ∠ X = 60°∠ Y = 90°∴∠ Z = 30°  . .....Angle sum property of a triangle i.e 180°Now appplying Trignometry identity we get[tex]\tan X = \frac{\textrm{side opposite to angle X}}{\textrm{side adjacent to angle X}}[/tex][tex]\tan 60 = \frac{YZ}{XY}\\\\\sqrt{3} =\frac{y}{3\sqrt{6} }\\  y= 3\sqrt{3} \sqrt{6} \\y= 9\sqrt{2}[/tex]Now,[tex]\sin X = \frac{\textrm{side opposite to angle C}}{Hypotenuse}\\\\\\\sin 60 = \frac{YZ}{XZ}\\ \\\frac{\sqrt{3} }{2} =\frac{9\sqrt{2} }{x} \\\\x=\frac{18\sqrt{2} }{\sqrt{3} } \\\textrm{after fationalizing the denominator root 3 we get}\\\\x=6\sqrt{6}[/tex]