In the given problem, we have a right triangle with a hypotenuse of length \( 10\sqrt{2} \) and an angle of \( 45^\circ \). However, there is a contradiction because for a triangle with a \( 45^\circ \) angle, both legs should be of equal length due to the properties of a \( 45^\circ-45^\circ-90^\circ \) triangle.
In a \( 45^\circ-45^\circ-90^\circ \) triangle, the relationship between the legs and the hypotenuse is given by:
\[ \text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}} \]
Let's check the length of the legs using the given hypotenuse:
\[ \text{leg} = \frac{10\sqrt{2}}{\sqrt{2}} = 10 \]
This suggests that both legs of the triangle should each be \( 10 \) units long.
However, the problem states that one of the sides opposite the \( 45^\circ \) angle is \( 14 \). This indicates that we are looking at another triangle configuration maybe stemming from a different interpretation or setup.
By exploring the situation further, the triangle with a side length of \( 14 \) and using the sine or cosine rules could involve a different triangle structure.
Given that the relevant angle remains \( 45^\circ \), and one side of the triangle is known. We need to assume or clarify whether this is another triangle somehow related to our original one.
To find the hypotenuse of the second triangle, where a leg is given:
Assuming this triangle is still a \( 45^\circ-45^\circ-90^\circ \) triangle as before:
If we take one leg \( b = 14 \),
The hypotenuse \( c \) for that triangle can be calculated as:
\[ c = b \cdot \sqrt{2} \]
Substituting \( b = 14 \):
\[ c = 14 \cdot \sqrt{2} \]
Thus, the length of the hypotenuse in the second triangle would be:
\[ \boxed{14\sqrt{2}} \]