In a 30-60-90 triangle, the ratios of the lengths of the sides are as follows:
- The side opposite the 30-degree angle (shortest side) is \(x\).
- The side opposite the 60-degree angle is \(x\sqrt{3}\).
- The hypotenuse (opposite the 90-degree angle) is \(2x\).
Given that the side opposite the 60-degree angle is \(63 - \sqrt{}\), let's denote it as \(y\).
According to the ratio, we have:
\[ y = x\sqrt{3} \]
Therefore, we can find \(x\) as:
\[ x = \frac{y}{\sqrt{3}} = \frac{63 - \sqrt{}}{\sqrt{3}} \]
Now, to find the hypotenuse \(h\):
\[ h = 2x = 2 \left(\frac{63 - \sqrt{}}{\sqrt{3}}\right) = \frac{2(63 - \sqrt{})}{\sqrt{3}} \]
Now, simplifying:
\[ h = \frac{126 - 2\sqrt{}}{\sqrt{3}} \]
To determine if we can match this with one of the provided answers, we need to plug-in the \(h\) back into the options. However, it seems there was ambiguity regarding the interpretation of \(63 - \sqrt{}\).
From the original options, we can check which one matches correctly:
- 12
- 6
- 23-√
- 123-√
If you decide \(63 - \sqrt{}\) to be a value rather than a variable calculation to see which options fit via calculations or ratio you might find a selected answer.
So upon evaluating the individual lengths, none of the fixed options seem to directly relate with the calculated hypotenuse unless \(y = 63 - 0\) which leads directly to a numerical calculation that can be transformed to either yield an interpretation of simplified ratios.
To conclude the hypotenuse you'll need no less than multiplying by \(2\), or starting back from individual selected simple references upto numerical assesment to ensure a correct match or generating \(h\).
For a numerical relationship leading from specific known angled values, select \( h = 12 \) if reflectively matching a simple geometric average around point estimates to extreme values deduced from distributions.
Assistant calculation would be needed based on clearer definitions about square representation or values of expression forming if one could clarify without square fiction seeming unanswered or mis-entered.
For concise conclusion, if specific numbers yield clear resolution options are left under simplified additive basis.
1 answer