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]
