Question

point b is on line segment \\(\overline{ac}\\). given \\(ac = 2x + 7\\), \\(bc = x\\), and \\(ab = 5x - 9\\), determine the numerical length of \\(\overline{ab}\\).

Explanation:

Step1: Apply segment addition postulate

Since B is on \( \overline{AC} \), we have \( AB + BC = AC \). Substitute the given expressions: \( (5x - 9) + x = 2x + 7 \).

Step2: Solve for x

Combine like terms: \( 6x - 9 = 2x + 7 \). Subtract \( 2x \) from both sides: \( 4x - 9 = 7 \). Add 9 to both sides: \( 4x = 16 \). Divide by 4: \( x = 4 \).

Step3: Find length of \( \overline{AB} \)

Substitute \( x = 4 \) into \( AB = 5x - 9 \): \( AB = 5(4) - 9 = 20 - 9 = 11 \).

Answer:

11