To find the length of the shortest leg \( x \) of the large triangle using the geometric mean, we need to apply the property of right triangles formed by a line drawn from the right angle vertex to the midpoint of the hypotenuse.
Given that the larger triangle has legs measuring \( 3 \) and \( 5 \), the geometric mean can be used to find the lengths of the segments created by the altitude drawn to the hypotenuse.
The segment lengths are \( a \) and \( b \), where \( a = 3 \) and \( b = 5 \).
The length of the altitude \( h \) to the hypotenuse can be given by the formula: \[ h = \sqrt{(3)(5)} = \sqrt{15} \]
Now, we can also visualize how this may relate to finding \( x \) directly as the shortest leg. If \( x \) is the length opposite the 3, we can use the relationship between the sides and the legs of the triangle using the formula for geometric means relative to triangles.
However, based on traditional solving approaches using geometric properties, here we can answer simply based on the options provided since the smaller leg can be verified against \( x \) values to estimate probable lengths by testing triangle inequalities quickly.
Testing each option in general sense not to rely solely on calculated approximation to assess basic integers, we isolate lengths proportional to existing numbers.
Out of the choices (2.8, 4.9, 8, and 15):
- Select \( 2.8 \) (non-integer small scope),
- Add the non-integer values leading toward inclusive \( <5 \),
- Distanced maximum integers lead us closer options.
With all context considered towards practical application to hypotenuse or triangle properties not directly leading to calculative properties indicated before, The correct answer here looking through these rounded values would focus on summing into a probable approximate value length related down towards:
The closest rounded value of \(2.8\) could be suggested by contextualizing shortest leg lengths to geometric means rounded in situational approximation would maximize.
Thus, based on this more practical perspective if drawn correctly from outlined properties, the final estimation for shortest leg size \( x \approx 2.8\).
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