In both right triangles, we can use the properties of the 45-45-90 triangle, where the angles of the triangle are 45 degrees, 45 degrees, and 90 degrees.
In a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse \( h \) can be determined using the relationship:
\[ h = \text{leg} \times \sqrt{2} \]
First Triangle:
- The side opposite the 45-degree angle is given as 10. Since this triangle should also have both legs equal, we find its hypotenuse:
\[ h_1 = 10 \times \sqrt{2} \]
However, since the hypotenuse length is also stated to be \( \frac{10}{2} = 5 \), it seems there might be confusion regarding the side lengths. But let’s move on to the second triangle.
Second Triangle:
- The side opposite the 45-degree angle is given as 14. Using the same relationship for the hypotenuse:
\[ h_2 = 14 \times \sqrt{2} \]
Now we calculate \( 14 \times \sqrt{2} \): \[ h_2 \approx 14 \times 1.414 \approx 19.8 \]
However, given the options listed, we should find the hypotenuse relative to the leg lengths provided in the options:
- Since the hypotenuse can be expressed as \( \text{leg} \times \sqrt{2} \), we consider comparing \( 14 \times \sqrt{2} \) against the options directly.
- The options are:
- a. \( \frac{7}{2} \)
- b. \( 7 \)
- c. \( 14 \)
- d. \( \frac{14}{2} = 7 \)
None of these values is equal to \( 14\sqrt{2} \), but observing option c, if we consider the leg length of 14 it could possibly fit.
However, with the direct conclusion from the above calc, the relevant expressions don't quite correspond. Reviewing each, the earlier conclusion cannot be tied against observed options without context.
Thus, it seems the expected calculations derive otherwise.
None particularly correctly reflects conventional derivation against the question. The hypotenuse indeed should maintain:
Conclusion from available choices None of those provided accurately aligns against \(14\sqrt{2}\); however, for leg to hypotenuse for provided leg \(14\) remains valid. Hence none retain soundness aligning entries into comparative ratioing definitively enough.
So let's summarize:
From accurate reflections regarding geometric properties, hypotenuse remains \(14\sqrt{2}\) as derived correctly not ultimately crystallizing under option presentation looks provided. But option c: 14 retains viable length if confounded appropriately might align if considered separate context from geometric base normalized.
Thus while reinterpretative on basis; confirm legit structure you'd warrant into assessments.