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To solve this problem, let's break it down based on the information provided in the triangle:

- The triangle is a right triangle, with one angle of \(30^\circ\) and another of \(60^\circ\).
- The side opposite the \(60^\circ\) angle is \(7\sqrt{3}\), and we are asked to find the hypotenuse \(x\).

### Using Trigonometric Ratios:
In a 30°-60°-90° triangle, the side relationships are known:
- The side opposite the \(30^\circ\) angle is half the length of the hypotenuse.
- The side opposite the \(60^\circ\) angle is \(\sqrt{3}\) times the length of the side opposite the \(30^\circ\) angle.

Given that the side opposite the \(60^\circ\) angle is \(7\sqrt{3}\), we can set up the ratio as:
\[
\text{Side opposite } 60^\circ = \sqrt{3} \times \text{Side opposite } 30^\circ
\]
So, the side opposite the \(30^\circ\) angle (let's call it \(a\)) is:
\[
7\sqrt{3} = \sqrt{3} \times a
\]
Dividing both sides by \(\sqrt{3}\):
\[
a = 7
\]
Now, the hypotenuse \(x\) is twice the side opposite the \(30^\circ\) angle:
\[
x = 2 \times 7 = 14
\]

### Verifying with Options:
It seems the answer \(14\) is not among the options. The problem might have a subtle trick or mistake in the provided options, or there may have been a different interpretation required. Nonetheless, based on the standard properties of a 30°-60°-90° triangle, the hypotenuse should be 14 units.

Would you like more details on any part of the calculation, or clarification on the options provided?

Here are 8 related questions for further exploration:
1. How are the side lengths of a 30°-60°-90° triangle related?
2. What are the properties of special right triangles?
3. How do you apply trigonometric ratios in right triangles?
4. What is the difference between solving triangles with and without given side lengths?
5. How can you solve for unknown sides using the Pythagorean theorem?
6. What is the sine function for a 30° angle?
7. How do you convert trigonometric results back into side lengths?
8. What strategies are effective for recognizing triangle properties in geometry problems?

**Tip:** Always double-check whether the problem is asking for exact or approximate values based on the provided options.

Learn how to find the value of x in a 30°-60°-90° triangle. This problem involves trigonometric ratios, specifically for a right triangle with angles 30°, 60°, and 90°. Detailed step-by-step solution included.

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