In a 30-60-90 triangle, the relationship between the lengths of the sides is as follows:
- The side opposite the 30-degree angle is the shortest and is usually denoted as \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse is \( 2x \).
Given that the side opposite the 60-degree angle is 3 cm long, we can set it equal to \( x\sqrt{3} \):
\[ x\sqrt{3} = 3 , \text{cm} \]
To find \( x \):
\[ x = \frac{3}{\sqrt{3}} = \sqrt{3} , \text{cm} \]
Now, we can find the hypotenuse:
\[ \text{Hypotenuse} = 2x = 2(\sqrt{3}) , \text{cm} \]
Now plugging in the value of \( x = \frac{3}{\sqrt{3}} \):
\[ \text{Hypotenuse} = 2\left(\frac{3}{\sqrt{3}}\right) = \frac{6}{\sqrt{3}} = 2\sqrt{3} \]
To find the numerical value, we know:
\[ \sqrt{3} \approx 1.732 \text{ so } 2\sqrt{3} \approx 2 \times 1.732 \approx 3.464 , \text{cm} \]
None of the options provided seem to be correct. However, if we strictly focus on the problem statement, the correct hypotenuse based on our work would be close to 6 cm or about 3.464 cm, and we need to state that the answer isn't directly provided in the options given.
If we deduce the appropriate value based on \( x \sqrt{3} \), the missing hypotenuse derived from the options would be relevant if we asserted \( 2x = 6 \) directly yielding \( x = 3 \).
This assumption leads us directly to state:
- For TV as assumed: 3 resulting in valid arcs interpreting either \( 6\sqrt{3} or 28 \) under rational scaling but remains incompatible to the presented.
Double check with \( 14 cm \) asking alongside root finding.
Any misunderstanding, clarifications requested.