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The problem asks to find the value of \( x \) in a \( 30^\circ - 60^\circ - 90^\circ \) triangle, where the side opposite the \( 30^\circ \) angle is given as 15 units.

In a \( 30^\circ - 60^\circ - 90^\circ \) triangle, the sides have the following ratio:
\[
1 : \sqrt{3} : 2
\]
The side opposite the \( 30^\circ \) angle is \( x \), the side opposite the \( 60^\circ \) angle is \( x\sqrt{3} \), and the hypotenuse is \( 2x \).

Given that the side opposite the \( 30^\circ \) angle is 15 (which corresponds to \( x \)), we can directly set:
\[
x = 15
\]

Now, let's verify if the side lengths match the ratios:
- Side opposite \( 60^\circ \) angle should be \( 15\sqrt{3} \).
- Hypotenuse should be \( 2(15) = 30 \).

Thus, the value of \( x \) is \( 15 \), and it is already simplified.

Would you like to explore the solution further or any related concepts? Here are some related questions:

1. What are the properties of a \( 30^\circ - 60^\circ - 90^\circ \) triangle?
2. How can the Pythagorean theorem be applied to this type of triangle?
3. What is the relationship between the angles and sides in a right triangle?
4. How would you find the length of the hypotenuse if the side opposite \( 60^\circ \) were known?
5. How do trigonometric functions like sine, cosine, and tangent relate to the sides of this triangle?

**Tip:** Always remember that in a \( 30^\circ - 60^\circ - 90^\circ \) triangle, the smallest side is half of the hypotenuse.

Find the value of x in the 30°-60°-90° triangle where the side opposite the 30° angle is 15. Give your answer as a simplified radical.

Learn how to find the value of x in a 30°-60°-90° triangle where the side opposite the 30° angle is 15. This problem covers key geometric concepts, including special right triangles and their side ratios. Suitable for students in Grades 9-11.

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