Question

1. if b is the midpoint of \\(overline{ac}\\), and \\(ac = 8x - 20\\), find \\(bc\\).

diagram: a---b---c with segment ab labeled \\(3x - 1\\)

Explanation:

Step1: Use midpoint property

Since \( B \) is the midpoint of \( \overline{AC} \), \( AB = BC \) and \( AC = AB + BC = 2AB \). Given \( AB = 3x - 1 \) and \( AC = 8x - 20 \), we have \( 8x - 20 = 2(3x - 1) \).

Step2: Solve for \( x \)

Expand the right side: \( 8x - 20 = 6x - 2 \). Subtract \( 6x \) from both sides: \( 2x - 20 = -2 \). Add 20 to both sides: \( 2x = 18 \). Divide by 2: \( x = 9 \).

Step3: Find \( BC \)

Since \( BC = AB = 3x - 1 \), substitute \( x = 9 \): \( BC = 3(9) - 1 = 27 - 1 = 26 \).

Answer:

\( 26 \)