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.