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finlep [7]
3 years ago
5

Answer both with STEPS

Mathematics
1 answer:
Amanda [17]3 years ago
8 0

Answer:

7 )

x =  \frac{3\sqrt{2} }{2}

y= 3

8 )

x=6\sqrt{6}

y= 9\sqrt{2}

Step-by-step explanation:

7  )                                 8)

In Δ ABC                                      In Δ XYZ            

∠ C = 45°                                          ∠ X = 60°

∠ A = 90°                                           ∠ Y = 90°

AC= \frac{3\sqrt{2} }{2}             XY= 3\sqrt{6}

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 =  \frac{3\sqrt{2} }{2}

Now appplying Trignometry identity we get

\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

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

\tan X = \frac{\textrm{side opposite to angle X}}{\textrm{side adjacent to angle X}}

\tan 60 = \frac{YZ}{XY}\\\\\sqrt{3} =\frac{y}{3\sqrt{6} }\\  y= 3\sqrt{3} \sqrt{6} \\y= 9\sqrt{2}

Now,

\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}

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