Question

point b is on line segment \\( \overline { a c } \\). given \\( a b = 4 x - 6 \\), \\( a c = 2 x + 10 \\), and \\( b c = 4 \\), determine the numerical length of \\( \overline { a c } \\).

Explanation:

Step1: Use segment addition postulate

Since \(B\) is on \(\overline{AC}\), \(AB + BC=AC\).
Substitute \(AB = 4x - 6\), \(AC = 2x + 10\), and \(BC = 4\) into the equation:
\(4x-6 + 4=2x + 10\)

Step2: Simplify the left - hand side of the equation

\(4x-2=2x + 10\)

Step3: Solve for \(x\)

Subtract \(2x\) from both sides: \(4x-2x-2=2x-2x + 10\), which gives \(2x-2 = 10\).
Add \(2\) to both sides: \(2x-2 + 2=10 + 2\), so \(2x=12\).
Divide both sides by \(2\): \(x=\frac{12}{2}=6\).

Step4: Find the length of \(AC\)

Substitute \(x = 6\) into \(AC=2x + 10\).
\(AC=2\times6+10=12 + 10=22\)

Answer:

\(22\)