We are given that:

• $AB = 2m + 10$
• $BC = x$
• $AC = 5m$

Since $B$ is a point on line segment $AC$, the total length $AC$ is the sum of the lengths of $AB$ and $BC$:

$AC = AB + BC$

Substituting the given values:

$5m = (2m + 10) + x$

Now, solve for $x$:

$5m = 2m + 10 + x$

Subtract $2m$ from both sides:

$3m = 10 + x$

Now, subtract 10 from both sides:

$x = 3m - 10$

Thus, the value of $BC$ (which is $x$) is $3m - 10$, and the total length of $AC$ is $5m$, as given.

Would you like further details or have any questions?

Here are 5 related questions:

1. What happens if the value of $m$ is given? How does that affect $BC$?
2. How do you check the consistency of the segment length using different values for $m$?
3. What are some common mistakes when setting up such equations?
4. If $AB$ and $BC$ were in terms of other variables, how would the approach change?
5. How does this method extend to other geometric figures?

Tip: Always double-check your equation setup by reviewing how the segments add up to the total length.