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The problem asks to find \( BC \) given the following conditions:

1. \( B \) is the midpoint of \( AC \).
2. \( AC = 8x - 20 \).
3. \( AB = 3x - 1 \).

### Step-by-step solution:

Since \( B \) is the midpoint of \( AC \), the length of \( AB \) and \( BC \) are equal. Therefore:
\[
AB = BC
\]
Additionally, the total length \( AC \) can be expressed as:
\[
AC = AB + BC = 2 \cdot AB
\]

Substitute the expression for \( AB \) into the equation for \( AC \):
\[
AC = 2 \cdot (3x - 1)
\]
Simplifying the right-hand side:
\[
AC = 6x - 2
\]

We are also given that \( AC = 8x - 20 \). So, we can set the two expressions for \( AC \) equal to each other:
\[
6x - 2 = 8x - 20
\]
Now, solve for \( x \):
\[
6x - 8x = -20 + 2
\]
\[
-2x = -18
\]
\[
x = 9
\]

### Find \( BC \):
Now that we have \( x = 9 \), substitute this value into the expression for \( AB \) (which is equal to \( BC \)):
\[
AB = BC = 3x - 1 = 3(9) - 1 = 27 - 1 = 26
\]

Thus, the length of \( BC \) is:
\[
BC = 26
\]

Would you like further details or explanations? Here are five related questions:

1. How would the solution change if \( B \) were not the midpoint of \( AC \)?
2. Can you find \( AB \) given different values for \( x \)?
3. What is the relationship between the midpoint and the total length in this problem?
4. How would you solve if \( AB \) and \( BC \) were given as different expressions?
5. How can the concept of midpoints be applied in geometry problems involving triangles?

**Tip:** Always check for consistency in midpoint problems by ensuring that the segments are equal when required.

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

Learn how to solve a linear equation where B is the midpoint of AC, AC = 8x - 20, and AB = 3x - 1. This problem involves key algebraic concepts like midpoints and linear equations. Suitable for Grade 8 students.

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