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Point [tex]$B$[/tex] lies along [tex]$\overline{AC}$[/tex] between points [tex]$A$[/tex] and [tex]$C$[/tex]. The length of [tex]$\overline{AC}$[/tex] is 38 centimeters. If [tex]$AB = 7x - 1$[/tex] and [tex]$BC = 4x + 6$[/tex], is [tex]$B$[/tex] the midpoint of [tex]$\overline{AC}$[/tex]? Explain your reasoning.

A. No, [tex]$B$[/tex] is not the midpoint of [tex]$\overline{AC}$[/tex]. Since [tex]$AB + BC = AC$[/tex], we can write the equation [tex]$(7x - 1) + (4x + 6) = 38$[/tex]. We can solve this equation to find that [tex]$x = 3$[/tex]. This means that [tex]$AB = 20$[/tex] cm and [tex]$BC = 18$[/tex] cm. [tex]$B$[/tex] is not the midpoint because [tex]$AB \neq BC$[/tex].

B. Yes, [tex]$B$[/tex] is the midpoint of [tex]$\overline{AC}$[/tex]. Since [tex]$AB + BC = AC$[/tex], we can write the equation [tex]$(7x - 1) + (4x + 6) = 38$[/tex]. We can solve this equation to find that [tex]$x = 3$[/tex]. This means that [tex]$AB = BC = 20$[/tex] cm. [tex]$B$[/tex] is the midpoint because [tex]$AB = BC$[/tex].

C. No, [tex]$B$[/tex] is not the midpoint of [tex]$\overline{AC}$[/tex]. Since [tex]$AB + BC = AC$[/tex], we can write the equation [tex]$(7x - 1) + (4x + 6) = 38$[/tex]. We can solve this equation to find that [tex]$x = 15$[/tex]. This means that [tex]$AB = 104$[/tex] cm and [tex]$BC = 66$[/tex] cm. [tex]$B$[/tex] is not the midpoint because [tex]$AB \neq BC$[/tex].

D. Yes, [tex]$B$[/tex] is the midpoint of [tex]$\overline{AC}$[/tex]. Since [tex]$AB + BC = AC$[/tex], we can write the equation [tex]$(7x - 1) + (4x + 6) = 38$[/tex]. We can solve this equation to find that [tex]$x = 15$[/tex]. This means that [tex]$AB = BC = 104$[/tex] cm. [tex]$B$[/tex] is the midpoint because [tex]$AB = BC$[/tex].



Answer :

GinnyAnswer
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.

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