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In this problem, we are given the following information:

- \( B \) is the midpoint of \( AC \).
- \( AC = 8x - 20 \).
- \( AB = 3x - 1 \).
- We are asked to find \( BC \).

### Step-by-Step Solution

Since \( B \) is the midpoint of \( AC \), it divides the line segment into two equal parts. This means:
\[
AB = BC
\]

Also, since \( AC = AB + BC \), we can express \( BC \) as \( BC = AB \). So:
\[
AC = 2 \times AB
\]

Substituting the values we know:
\[
8x - 20 = 2 \times (3x - 1)
\]

### Solve for \( x \):
\[
8x - 20 = 6x - 2
\]

Now, simplify:
\[
8x - 6x = -2 + 20
\]
\[
2x = 18
\]
\[
x = 9
\]

### Find \( BC \):
Since \( AB = BC \) and \( AB = 3x - 1 \), substitute \( x = 9 \) into the expression for \( AB \):
\[
AB = 3(9) - 1 = 27 - 1 = 26
\]
So, \( BC = 26 \).

### Final Answer:
\[
BC = 26
\]

Would you like more details or have any further questions?

Here are five related questions for further practice:
1. If \( AC = 10x - 15 \) and \( AB = 4x + 3 \), how do you find \( BC \)?
2. What happens to the length of \( BC \) if \( B \) is no longer the midpoint of \( AC \)?
3. How would the equation change if \( AB \) and \( BC \) were in a ratio of 2:1?
4. If \( AC \) was expressed as a quadratic function, how would you solve for the length of \( BC \)?
5. What if \( AB \) was a function of another variable, say \( y \)? How would you approach the solution?

**Tip**: Always check midpoint problems by confirming that \( AB = BC \), which ensures your equation setup is correct.

If B is the midpoint of AC, and AC = 8x - 20, find BC. AB = 3x - 1.

Learn how to find BC when B is the midpoint of AC. The problem involves algebraic expressions and the Midpoint Theorem. The equation AC = 8x - 20 and AB = 3x - 1 are used to calculate the value of BC. This solution is suitable for students in Grades 8-10.

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