The problem asks to identify the hypotenuse of the right triangle shown in the image.

1. Problem Description

The problem asks to identify the hypotenuse of the right triangle shown in the image.

2. Solution Steps

The hypotenuse of a right triangle is the side opposite the right angle. In the given triangle, the right angle is at vertex $X$. Therefore, the side opposite to the right angle $X$ is the segment connecting vertices $E$ and $G$, which is segment $\overline{EG}$.

3. Final Answer

A. $\overline{EG}$