Question

1. if b is the midpoint of \\(\overline{ac}\\), and \\(ac = 8x - 20\\), find \\(bc\\). (diagram: segment \\(ac\\) with midpoint \\(b\\), \\(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: \( 2x - 20 = -2 \). Add 20 to both sides: \( 2x = 18 \). Divide by 2: \( x = 9 \).

Step3: Find BC (which equals AB)

Substitute \( x = 9 \) into \( AB = 3x - 1 \): \( 3(9) - 1 = 27 - 1 = 26 \). So \( BC = 26 \).

Answer:

26