This is a 30°-60°-90° triangle, which has specific side length ratios. In such a triangle:

1. The side opposite the 30° angle (shorter leg) is half the hypotenuse.
2. The side opposite the 60° angle (longer leg) is the shorter leg multiplied by $\sqrt{3}$.

Given:

• Hypotenuse = 9
• We need to find $x$, the side opposite the 60° angle.

Step-by-step solution:

1. Shorter leg (opposite 30°): $\text{Shorter leg} = \frac{\text{Hypotenuse}}{2} = \frac{9}{2}.$

2. Longer leg (opposite 60°): $x = \text{Shorter leg} \cdot \sqrt{3} = \frac{9}{2} \cdot \sqrt{3} = \frac{9\sqrt{3}}{2}.$

Final Answer:

$x = \frac{9\sqrt{3}}{2}$

Let me know if you'd like a detailed explanation of any step or have further questions!

Related Questions:

1. What is the ratio of sides in a 30°-60°-90° triangle, and why does it work?
2. How can you derive the side lengths of any special triangle?
3. What would the hypotenuse be if $x = \frac{12\sqrt{3}}{2}$?
4. How can you rationalize a denominator when it contains $\sqrt{3}$?
5. What other special triangles exist, and how do their ratios differ?

Tip: For 30°-60°-90° triangles, always start by identifying the hypotenuse and shorter leg, then use the known ratios to find the other sides!