Question

7.Find the lengths of the missing sides in the triangle. If your answer is not an integer, leave it in simplest radical form. The diagram is not drawn to scale.

Answer 1

Here a right angled triangle given. one angle with measure
`45^o` given. The three sides of the triangle given 4, x, y.

We have to find, the sides which is opposite, adjacent and hypotenuse here.

We know that the side opposite to right angle is always hypotenuse. So, hypotenuyse = y.

The side adjacent to the given angle
`45^o` is x. So, here adjacent = x.

The opposite side is opposite to the given angle. So, opposite = 4.

Now we will use SOHCAHTOA that is
`sin(x) =(Opposite)/(Hypotenuse) , cos(x) =(Adjacent)/(Hypotenuse) , tan(x) =(Opposite)/(Adjacent)`, where x is the angle given.

To get x, we will use tan. So we will get,

`tan(45^o) = (4)/(x)`

We know the value of
`tan(45^o) = 1`. By substituting the value we will get,

`1 =(4)/(x)`

To find x, we have to move x here to the left side by multiplying it to both sides. We will get,

`(1)(x) = ((4)/(x)) (x)`

`x = 4`

So we have got the value of x here.

Now to find y, we will use the trigonometric function sine.

`sin(45^o) =(4)/(y)`

we know the value of
`sin(45^o) =(√(2))/(2)`

By substituting the value we will get,

`(√(2))/(2) = (4)/(y)`

By cross multiplying we will get,

`(√(2)) (y) = (4)(2)`

`√(2)y = 8`

We will get y by dividing both sides by
`√(2)`, we will get,

`(√(2)y)/(√(2)) =(8)/(√(2) )`

`y =(8)/(√(2) )`

Now we will rationalize the denominator by multiplying
`√(2)` to the top and bottom.

`y =(8√(2))/((√(2))(√(2)))`

`y =(8√(2))/(2)`

`y = 4√(2)`

So we have got the required values of x and y.

Answer 2

x = 4 / tan(45)
x = 4

y = 4 / sin(45)
y = 4 sqrt(2)