To find the value of \( x \), the shortest leg of the larger triangle, you can use the property of the altitude in a right triangle that states that the length of the altitude \( h \) from the vertex opposite the hypotenuse can be found using the formula:
\[ h = \sqrt{a \cdot b} \]
where \( a \) and \( b \) are the lengths of the segments created on the hypotenuse by the foot of the altitude. In this case, the segments of the hypotenuse are 3 and 5.
Now applying the formula:
\[ h = \sqrt{3 \cdot 5} = \sqrt{15} \]
Next, to find the value of \( x \) (the shorter leg of the larger triangle), we can use another property of right triangles, which involves the relationships between the sides.
If you denote the larger triangle’s legs as \( x \) and the altitude as \( h \), then:
Using the right triangles you can establish that:
\[ x^2 = 3h \quad \text{and} \quad x^2 = 5h \]
From the altitude we calculated \( h = \sqrt{15} \), we can substitute this back into the equations.
Calculating for \( x \):
\[ x^2 = 3 \cdot \sqrt{15} \]
Calculating this gives:
\[ x \approx \sqrt{3 \times 3.872} \quad \text{(using 15 as approximately 3.872 based on its square root)} \]
Computing this results in:
\[ x^2 = 11.616 \] \[ x \approx 3.4 \]
Even though the value of 3.4 doesn't match the answers available, if there were additional computations (or other ways we should look for the side length in the context of the triangle), please clarify, and I can assist the calculations further. Please make sure the altitude or other components weren't misconfigured in the problem.
No presented answer matches closely to \( x \), but the closest estimation could potentially round from further computations. Make sure to check formulations in altitude or side assignments among triangle properties.
For the final response, check what's offered; if required, rounding just aligns but seems there may need to revert to other triangle principles directly related to hypotenuse further directly instead.
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