In the image, we are asked to find the length of $GI$ in the square $FGHI$. The point $E$ is inside the square, and it appears that $E$ is connected to the vertices $F$, $G$, $H$, and $I$. One of the given lengths is $GE = 19$.

To solve for $GI$, here are a few observations:

• Since $FGHI$ is a square, all sides of the square are equal in length.
• The diagonals of a square are equal and divide the square into four right triangles.
• In a square, the diagonals are related to the side length $s$ by the formula: $d = s\sqrt{2}$, where $d$ is the diagonal length.

However, the problem does not directly provide the side length or diagonal length. But based on symmetry and geometric properties, we can infer the solution given additional details or simplifications about $E$'s role in the problem. It appears that $GI$ might be deduced based on symmetry or triangle properties formed within the square.

Could you provide more details about point $E$'s positioning or any other information that might help? Alternatively, I can calculate a solution based on assumptions about symmetry.

Let me know if you have further questions or if you'd like me to explore the problem differently!

Here are 5 related questions:

1. How do the diagonals of a square relate to its side length?
2. What are the properties of the right triangles formed by the diagonals of a square?
3. How do you calculate the length of a diagonal in any square?
4. Can the Pythagorean Theorem help in solving for side lengths within squares?
5. What role does symmetry play in solving problems involving squares?

Tip: In geometry problems involving squares, always explore the properties of diagonals and the triangles they form to simplify calculations.