Question

Answer both with STEPS

Answer 1

7 )

x =
`(3√(2) )/(2)`

`y= 3`

8 )

`x=6√(6)`

`y= 9√(2)`

Explanation:

7 ) 8)

In Δ ABC In Δ XYZ

∠ C = 45° ∠ X = 60°

∠ A = 90° ∠ Y = 90°

`AC= (3√(2) )/(2)`
`XY= 3√(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 =
`(3√(2) )/(2)`

Now appplying Trignometry identity we get

`\sin C = \frac{\textrm{side opposite to angle C}}{Hypotenuse}\\\\\sin 45 = (AC)/(BC)\\\\(1)/(√(2) ) =((3√(2) )/(2))/(y)\\\\y=(3* √(2)* √(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 = (YZ)/(XY)\\\\√(3) =(y)/(3√(6) )\\  y= 3√(3) √(6) \\y= 9√(2)`

Now,

`\sin X = \frac{\textrm{side opposite to angle C}}{Hypotenuse}\\\\\\\sin 60 = (YZ)/(XZ)\\ \\(√(3) )/(2) =(9√(2) )/(x) \\\\x=(18√(2) )/(√(3) ) \\\textrm{after fationalizing the denominator root 3 we get}\\\\x=6√(6)`