Question

Points A, B, and C are collinear, and B is the midpoint. If AC = 5x – 8 and AB = 2x + 2, what is BC?

Answer 1

Since points A, B, and C are collinear and B is the midpoint of segment AC, it follows that \( AB = BC \).

Given:
- \( AC = 5x - 8 \)
- \( AB = 2x + 2 \)

Knowing that \( AB + BC = AC \) and since \( AB = BC \), we can write:

\[
AB + AB = AC
\]
\[
2AB = AC
\]

Substituting \( AB \) and \( AC \) into the equation, we get:

\[
2(2x + 2) = 5x - 8
\]

Now, expand the left side:

\[
4x + 4 = 5x - 8
\]

Next, rearranging gives:

\[
4 + 8 = 5x - 4x
\]
\[
12 = x
\]

Now, substitute \( x = 12 \) back into the expression for \( AB \):

\[
AB = 2x + 2 = 2(12) + 2 = 24 + 2 = 26
\]

Thus, since \( AB = BC \):

\[
BC = 26
\]

Finally, the value of \( BC \) is:

\[
\boxed{26}
\]