Question

congruence stations if b is the midpoint of \\(\overline{ac}\\), and \\(ac = 8x - 20\\), find \\(bc\\). line segment with points a, b, c; segment ab labeled \\(3x - 1\\)

Explanation:

Step1: Use midpoint property

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

Step2: Solve for \( x \)

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

Step3: Find \( AB \) then \( BC \)

Substitute \( x = 9 \) into \( AB = 3x - 1 \): \( AB=3\times9 - 1=27 - 1 = 26 \). Since \( BC = AB \) (midpoint), \( BC = 26 \).

Answer:

\( 26 \)