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In a 30°-60°-90° triangle, the side lengths follow a specific ratio: the sides are in the ratio of \(1 : \sqrt{3} : 2\), where:

- The shortest side (opposite the 30° angle) is \(x\),
- The side opposite the 60° angle is \(x\sqrt{3}\),
- The hypotenuse (opposite the 90° angle) is \(2x\).

Let’s examine each answer choice to determine which one does not fit this ratio:

- **Option (a)**: \( 9, \frac{9}{2}, 9\sqrt{3} \)
  - If \(x = 9\), then the sides should be \(9, 9\sqrt{3}, 18\). However, \(\frac{9}{2}\) does not match this ratio. Therefore, this set is **incorrect** for a 30°-60°-90° triangle.
  
- **Option (b)**: \( 4, 8, 4\sqrt{3} \)
  - Here, if \(x = 4\), the sides should be \(4, 4\sqrt{3}, 8\), which fits the \(1 : \sqrt{3} : 2\) ratio. This set is **correct** for a 30°-60°-90° triangle.

- **Option (c)**: \( 7, 14, 7\sqrt{3} \)
  - If \(x = 7\), the sides should be \(7, 7\sqrt{3}, 14\), which fits the ratio perfectly. This set is **correct** for a 30°-60°-90° triangle.

- **Option (d)**: \( \frac{5}{3}, \frac{10}{3}, \frac{5\sqrt{3}}{3} \)
  - If \(x = \frac{5}{3}\), the sides should be \(\frac{5}{3}, \frac{5\sqrt{3}}{3}, \frac{10}{3}\), which fits the ratio. This set is also **correct** for a 30°-60°-90° triangle.

### Conclusion:
The set that **cannot** be the lengths of a 30°-60°-90° triangle is **option (a)**: \(9, \frac{9}{2}, 9\sqrt{3}\), as \(\frac{9}{2}\) does not follow the required ratio.

Let me know if you need more details or have any other questions!

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### Related questions:
1. How can you derive the side length ratios for a 30°-60°-90° triangle?
2. What is the relationship between the angles and sides in a right triangle?
3. How do you verify whether a set of numbers forms a right triangle?
4. Can any set of three numbers form a right triangle? Why or why not?
5. What are some real-life applications of the 30°-60°-90° triangle?

### Tip:
Always remember that the 30°-60°-90° triangle has a consistent ratio of \(1 : \sqrt{3} : 2\), which can help you quickly verify if a set of lengths is valid.

Which of the following is the only set of measures that CANNOT be the lengths of a 30°-60°-90° triangle?

Learn how to determine which sets of side lengths can form a 30°-60°-90° triangle. Explore the ratio 1:√3:2 that governs the sides of this special right triangle. This problem is suitable for high school students in Grades 9-11 and involves key concepts in geometry and trigonometry.

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