In a right triangle, the geometric mean of the two legs can be used to find the length of the third leg when you have two known legs. The formula for the geometric mean \( x \) of two legs \( a \) and \( b \) relates to the third leg \( c \) as follows:
\[ x^2 = a \cdot b \]
In your case, the two short legs are \( 3 \) and \( 5 \), and we want to find \( x \) such that:
\[ x^2 = 3 \cdot 5 \]
Calculating this gives:
\[ x^2 = 15 \]
Now take the square root to find \( x \):
\[ x = \sqrt{15} \]
Calculating \( \sqrt{15} \):
\[ \sqrt{15} \approx 3.872983346 \]
Rounding to the nearest tenth, we find:
\[ x \approx 3.9 \]
Since \(3.9\) is not one of the options you provided, it seems like there’s a misunderstanding with the context of the problem, or the options do not correspond correctly to the geometric relationships. Make sure to double-check the original problem statement to clarify the relationships among the lengths. If you are sure about the legs of \(3\) and \(5\) and their relationship in determining \(x\), the derived value from the calculations is \(3.9\).
If assuming \(x\) is an included leg in a different triangle setup where it relates to either \(3\) or \(5\) in a different positional analysis, a visual or contextual re-evaluation may be necessary.
Please ensure there are no additional context or constraints in the original problem that might alter the approach.
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