In a 30-60-90 triangle, the sides are in a specific ratio based on the angles. The side opposite the 30-degree angle is \(x\), the side opposite the 60-degree angle is \(x\sqrt{3}\), and the hypotenuse is \(2x\).
In this case, the side opposite the 60-degree angle is given as \(63 - \sqrt{1}\).
To find the value of \(x\), we can set up the equation: \[ x\sqrt{3} = 63 - \sqrt{1} \]
Since \(\sqrt{1} = 1\), we can rewrite it as: \[ x\sqrt{3} = 63 - 1 \] \[ x\sqrt{3} = 62 \]
Now, we can solve for \(x\): \[ x = \frac{62}{\sqrt{3}} = \frac{62\sqrt{3}}{3} \]
The hypotenuse \(c\) is given by: \[ c = 2x = 2 \times \frac{62\sqrt{3}}{3} = \frac{124\sqrt{3}}{3} \]
Now, let's simplify this further: We see that the response options do not align with this answer as they are written in different formats. Let's look through the choices once again and adapt to the math:
If we redo the equation setup as follows from the side opposite the 60 degree being directly \(63\):
- Recognize the \(x = 63/\sqrt{3}\)
- The hypotenuse is \(2x\): \[ hypotenuse = 2 \times \frac{63}{\sqrt{3}} = \frac{126}{\sqrt{3}} = 42\sqrt{3} \]
This does not yield consistent interpretation whether from transcription or computation based on pure results.
Thus amongst provided options, none actually represent a solution derived through correction or would represent transformation for resolving \(x\) and hypotenuse.
Normally, a hypotenuse of these would lean to common numeric combinations based off primary mathematics to fit. So as such, the length of the hypotenuse can't distinctly match your listed responses directly without a referential framework.
Therefore, there might involve needs here for cross-verifying understanding or simply requiring a formatting approach against answers.
Please address if further assistance is required, or re-evaluating the initial question setups may arise clarity.