In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. Specifically, the side opposite the 30-degree angle (the shortest side) is half the length of the hypotenuse.
Given that the side opposite the 60-degree angle is \( 63 - \sqrt{} \), we can set this length equal to \( x \sqrt{3} \), where \( x \) is the length of the side opposite the 30-degree angle.
Let's use the properties of the triangle:
- The side opposite the 60-degree angle: \( x \sqrt{3} \)
- The side opposite the 30-degree angle: \( x \)
- The hypotenuse: \( 2x \)
Since we know the side opposite the 60-degree angle equals \( 63 - \sqrt{} \), we can relate it to the ratios:
\[ x \sqrt{3} = 63 - \sqrt{} \]
To find \( x \) (the side opposite the 30-degree angle), we can express \( x \) in terms of the side opposite the 60-degree angle:
\[ x = \frac{63 - \sqrt{}}{\sqrt{3}} \]
Now the hypotenuse \( h = 2x \):
\[ h = 2 \left(\frac{63 - \sqrt{}}{\sqrt{3}}\right) = \frac{2(63 - \sqrt{)}}{\sqrt{3}} = \frac{126 - 2\sqrt{}}{\sqrt{3}} \]
Since the answers you provided don’t appear to match this expression directly, let's check the options:
- 12
- \( 12\sqrt{3} \)
- \( 6 \)
- \( 12\sqrt{3} \)
If we assume \( 63 - \sqrt{} = 12\sqrt{3} \), we can derive the lengths.
For a 30-60-90 triangle, if the side opposite 60 is \( x \sqrt{3} \) then the hypotenuse is:
If \( x \) being 12, it would give reconcile the known options. So, let’s validate the given length and options:
- Side opposite \( 30^\circ \) = \( 12 \)
- Hypotenuse = \( 2 \cdot 12 = 24 \)
- But doesn’t match options.
To check if it’s right, see if:
Hypotenuse \( = 12\sqrt{3} \)
Confirming you can’t have a length longer than the side provided.
The accurate value appears to align accurately to a shorter side when verified against hypotenuses.
Conclusively, it appears we miscalculate, since outputs dictate answers lean towards:
The hypotenuse length here from options can reasonably be summarized close to \( 12 \sqrt{3} \) assuming a numerical check.
Thus final is: 12 sqrt 3.