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Find the value of [tex]$x[tex]$[/tex] if [tex]$[/tex]A, B[tex]$[/tex], and [tex]$[/tex]C[tex]$[/tex] are collinear points and [tex]$[/tex]B[tex]$[/tex] is between [tex]$[/tex]A[tex]$[/tex] and [tex]$[/tex]C$[/tex].

[tex]AB = 3x[/tex]
[tex]BC = 2x - 7[/tex]
[tex]AC = 2x + 35[/tex]

A. 12
B. 7
C. 9
D. 14



Answer :

GinnyAnswer
To determine the value of \( x \), given the points \( A \), \( B \), and \( C \) are collinear and that \( B \) lies between \( A \) and \( C \), we start with the given measurements:

[tex]\[ AB = 3x \][/tex]
[tex]\[ BC = 2x - 7 \][/tex]
[tex]\[ AC = 2x + 35 \][/tex]

Since point \( B \) is between \( A \) and \( C \), the sum of the segments \( AB \) and \( BC \) should equal the total segment \( AC \). Therefore, we can set up the following equation:

[tex]\[ AB + BC = AC \][/tex]

Substituting the given expressions for \( AB \), \( BC \), and \( AC \):

[tex]\[ 3x + (2x - 7) = 2x + 35 \][/tex]

Now, we combine like terms on the left side of the equation:

[tex]\[ 3x + 2x - 7 = 2x + 35 \][/tex]

This simplifies to:

[tex]\[ 5x - 7 = 2x + 35 \][/tex]

Next, to isolate \( x \), we subtract \( 2x \) from both sides of the equation:

[tex]\[ 5x - 2x - 7 = 35 \][/tex]

This further simplifies to:

[tex]\[ 3x - 7 = 35 \][/tex]

Next, we add 7 to both sides of the equation to isolate the term with \( x \):

[tex]\[ 3x = 42 \][/tex]

Finally, we divide both sides by 3 to solve for \( x \):

[tex]\[ x = \frac{42}{3} = 14 \][/tex]

Therefore, the value of \( x \) is

[tex]\[ \boxed{14} \][/tex]
So, the correct answer is:

[tex]\[ \text{D. 14} \][/tex]

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