In a 30-60-90 triangle, the lengths of the sides are in a specific ratio:
- The side opposite the 30-degree angle is \( x \).
- The side opposite the 60-degree angle is \( x\sqrt{3} \).
- The hypotenuse (the side opposite the 90-degree angle) is \( 2x \).
Given that the side opposite the 60-degree angle is \( 63 - \sqrt{1} \), we can denote this length as \( x\sqrt{3} = 63 - \sqrt{1} \).
To find \( x \): \[ x = \frac{63 - \sqrt{1}}{\sqrt{3}}. \]
Now, to find the hypotenuse, we use the hypotenuse formula: \[ \text{Hypotenuse} = 2x = 2 \left( \frac{63 - \sqrt{1}}{\sqrt{3}} \right) = \frac{2(63 - \sqrt{1})}{\sqrt{3}}. \]
To summarize, the hypotenuse is \( \frac{126 - 2\sqrt{1}}{\sqrt{3}} \).
Now let's find the numerical approximation for the hypotenuse using a calculator:
- Substitute \(\sqrt{1}\) with \(1\).
- Calculate \(126 - 2\): \[ \frac{126 - 2}{\sqrt{3}} = \frac{124}{\sqrt{3}}. \]
- To simplify, we can rationalize: \[ \frac{124\sqrt{3}}{3}. \]
This indicates that the overall length of the hypotenuse is more complex than the original response options. Upon checking them again, none of the five options directly matches this expression without the need for approximations.
You may want to check the other lengths carefully again like actual numerical values since specific simplifications might inadvertently hint at one of the options above without context or initial direct computation. The original lengths were thrown as tricky inputs that might not be straightforward, but the underlying formula and aspects depict ways to gather the hypotenuse affirmed values straightforwardly.
If you’re still looking for the direct match to the choice, please clarify further complementary context, but overall working through does yield the hypotenuse's aspect to follow through.