Answer: We can use the trigonometric ratios of a 30-60-90-degree triangle to find the length of side x. In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle.
Let's call the length of the hypotenuse h. Then, the length of the side opposite the 30-degree angle is h/2, and the length of the side opposite the 60-degree angle is (h/2)√3.
Using this information, we can set up the following equation:
(h/2)√3 = 9
Solving for h, we get:
h = 18/√3
To find the length of side x, we need to use the trigonometric ratio for the sine of the 60-degree angle:
sin(60°) = opposite/hypotenuse
sin(60°) = x/(h/2)
Substituting h = 18/√3, we get:
sin(60°) = x/((18/√3)/2)
Simplifying, we get:
sin(60°) = x/(9/√3)
√3/2 = x/(9/√3)
Multiplying both sides by 9/√3, we get:
x = (9/√3) * (√3/2)
Simplifying, we get:
x = (9/2)√3
Therefore, the length of side x is (9/2)√3 with a rational denominator.
Explanation: