To determine if point [tex]\( B \)[/tex] is the midpoint of [tex]\(\overline{A C}\)[/tex], we will begin by using the information given and performing step-by-step calculations.
1. We're given the length of [tex]\(\overline{A C} = 38\)[/tex] centimeters.
2. Point [tex]\( B \)[/tex] lies between points [tex]\( A \)[/tex] and [tex]\( C \)[/tex], such that [tex]\( \overline{A B} = 7x - 1 \)[/tex] and [tex]\( \overline{B C} = 4x + 6 \)[/tex].
3. We know from the segment addition postulate that:
[tex]\[
AB + BC = AC
\][/tex]
4. Plugging in the given expressions, this becomes:
[tex]\[
(7x - 1) + (4x + 6) = 38
\][/tex]
5. Simplify the equation:
[tex]\[
7x - 1 + 4x + 6 = 38
\][/tex]
[tex]\[
11x + 5 = 38
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
11x + 5 = 38
\][/tex]
[tex]\[
11x = 33
\][/tex]
[tex]\[
x = 3
\][/tex]
7. Now that we have [tex]\( x = 3 \)[/tex], we can calculate the lengths of [tex]\( \overline{A B} \)[/tex] and [tex]\( \overline{B C} \)[/tex]:
[tex]\[
AB = 7x - 1 = 7(3) - 1 = 21 - 1 = 20 \text{ cm}
\][/tex]
[tex]\[
BC = 4x + 6 = 4(3) + 6 = 12 + 6 = 18 \text{ cm}
\][/tex]
8. For [tex]\( B \)[/tex] to be the midpoint of [tex]\(\overline{A C}\)[/tex], [tex]\( \overline{A B} \)[/tex] must equal [tex]\( \overline{B C} \)[/tex]. Comparing the lengths:
[tex]\[
AB = 20 \text{ cm}, \quad BC = 18 \text{ cm}
\][/tex]
[tex]\[
AB \neq BC
\][/tex]
Therefore, since [tex]\( AB \)[/tex] is not equal to [tex]\( BC \)[/tex], point [tex]\( B \)[/tex] is not the midpoint of [tex]\(\overline{A C}\)[/tex]. The correct reasoning and steps lead us to conclude that [tex]\( B \)[/tex] is not the midpoint.
