To determine the length of the line segment $FG$, we can use the fact that $F$ is on line segment $EG$. The total length of $EG$ is the sum of the lengths of $EF$ and $FG$.

Given:

• $EG = 19$
• $EF = 15$

We want to find $FG$.

Since $EG = EF + FG$, we can substitute the given values into the equation:

$19 = 15 + FG$

To find $FG$, subtract $15$ from both sides:

$FG = 19 - 15 = 4$

So, the length of $FG$ is $4$ units.

Would you like more details or have any other questions? Here are some related questions you might find useful:

1. If $F$ is the midpoint of $EG$, what would the lengths of $EF$ and $FG$ be?
2. How would you calculate $FG$ if you only knew the coordinates of points $E$, $F$, and $G$?
3. If $EF = 15$ and $FG = 2x + 3$, find $x$ when $EG = 19$.
4. What happens to the length of $FG$ if point $F$ moves closer to point $G$?
5. If $F$ divides $EG$ in the ratio $3:1$, what are the lengths of $EF$ and $FG$?

Tip: When dealing with line segments and points on a line, breaking the problem down into smaller parts, as we did here, can simplify the calculations significantly.