Given the problem where $Y$ is between $X$ and $Z$, with $XY = 2x + 1$, $YZ = 6x$, and $XZ = 81$, we can use the relationship:

$XY + YZ = XZ$

Substitute the given values:

$(2x + 1) + 6x = 81$

Combine like terms:

$8x + 1 = 81$

Subtract 1 from both sides:

$8x = 80$

Divide by 8:

$x = 10$

Now, substitute $x = 10$ back into the expression for $XY$:

$XY = 2x + 1 = 2(10) + 1 = 20 + 1 = 21$

Answer: The length of $XY$ is 21.

Would you like more details or have any questions?

Here are five related questions:

1. What is the length of $YZ$ with the calculated value of $x$?
2. If $XZ$ were a different value, how would you adjust the calculations?
3. What does it mean geometrically for $Y$ to be between $X$ and $Z$?
4. How do you solve for $x$ when given equations involving segments?
5. What would happen if $Y$ was not exactly between $X$ and $Z$ in terms of segment addition?

Tip: Always double-check the equation setup to ensure that all segments are accounted for correctly when applying the segment addition postulate.