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 is \( 2x \).
In this case, the side opposite the 60-degree angle is given as \( 63\sqrt{2} \).
Let’s denote the side opposite the 30-degree angle as \( x \): \[ x\sqrt{3} = 63\sqrt{2} \] To find \( x \), we can rearrange this: \[ x = \frac{63\sqrt{2}}{\sqrt{3}} \]
Now, we can rationalize the denominator: \[ x = 63 \cdot \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = 63 \cdot \frac{\sqrt{6}}{3} = 21\sqrt{6} \]
Now we can find the length of the hypotenuse, which is \( 2x \): \[ \text{Hypotenuse} = 2x = 2(21\sqrt{6}) = 42\sqrt{6} \]
Since \( 42\sqrt{6} \) does not seem to be one of the options provided, let’s double-check what was given: If the side opposite the 60-degree angle is \( 63-\sqrt{?} \) with a specific value, we could potentially work with that.
However, it seems my calculation based on the side values seems firm. Still, if additional context like specific numerical values was available, let us know.
If we inspect \( 63-\sqrt{?} \) specifically or clarify any given parameters, we could attempt another check accordingly. Please verify the \(63√{}\) claim or context to ensure it is correct.