Solve for x in a 30-60-90 Right Triangle

The image shows a right-angled triangle with angles , , and . This is a special type of right-angled triangle known as a 30-60-90 triangle.

In a 30-60-90 triangle, the sides have a specific ratio. Let the side opposite the angle be . Then the side opposite the angle is , and the side opposite the angle (the hypotenuse) is .

In the given triangle:
The angle opposite the side with length 9 is .
The angle opposite the side with length is .
The angle opposite the hypotenuse is .

Using the properties of a 30-60-90 triangle, we can set up the following relationships:
Let be the side opposite the angle. So, .
The side opposite the angle is . We are given that this side has length 9. So, .
The hypotenuse is .

We can use the equation to find the value of .
Divide both sides by :

To rationalize the denominator, multiply the numerator and denominator by :

Since , we have .

Alternatively, we can use trigonometric ratios.
We know that .
For the angle , the opposite side is and the adjacent side is 9.
So, .
We know that .
Therefore, .
Multiply both sides by 9:

Rationalize the denominator:
.

For the angle , the opposite side is 9 and the adjacent side is .
So, .
We know that .
Therefore, .
Multiply both sides by :

Divide both sides by :

Rationalize the denominator:
.

The final answer is .