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We are given a right triangle with angles 30°, 60°, and 90°. The side opposite the 60° angle is labeled 9, and we are tasked with finding the length of side \( x \), which is opposite the 30° angle, in simplest radical form.

This triangle is a 30°-60°-90° triangle, which has a known ratio of side lengths:
- The side opposite the 30° angle is \( \frac{1}{2} \) the hypotenuse.
- The side opposite the 60° angle is \( \frac{\sqrt{3}}{2} \) the hypotenuse.

In this case:
- The side opposite the 60° angle (9) is \( \frac{\sqrt{3}}{2} \) the hypotenuse.
So, let the hypotenuse be \( h \). We can write:
\[
\frac{\sqrt{3}}{2}h = 9
\]
Solving for \( h \):
\[
h = \frac{9 \times 2}{\sqrt{3}} = \frac{18}{\sqrt{3}} = \frac{18\sqrt{3}}{3} = 6\sqrt{3}
\]

Now, to find the side opposite the 30° angle (which is \( x \)):
\[
x = \frac{1}{2}h = \frac{1}{2} \times 6\sqrt{3} = 3\sqrt{3}
\]

Thus, the length of side \( x \) in simplest radical form is \( 3\sqrt{3} \).

Would you like further clarification on any part of the solution? Here are some related questions for more practice:

1. How do you derive the side-length ratios in a 30°-60°-90° triangle?
2. What is the relationship between the side lengths of a 45°-45°-90° triangle?
3. How can you apply the Pythagorean theorem to check the results in this type of triangle?
4. What would the value of \( x \) be if the side opposite the 60° angle was 12 instead of 9?
5. How do trigonometric functions (sine, cosine, tangent) apply to solve for unknown sides in non-special right triangles?

**Tip**: Always recognize the patterns of special triangles, such as 30°-60°-90°, to quickly solve problems involving unknown side lengths.

Find the length of side x in simplest radical form with a rational denominator in a 30°-60°-90° triangle, where the side opposite the 60° angle is 9.

Solve a 30°-60°-90° triangle where the side opposite the 60° angle is 9. Learn how to find the length of side x in simplest radical form with a rational denominator using known ratios.

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