Question

1. if b is the mid - point of $overline{ac}$, and $ac = 8x - 20$, find bc.

$3x - 1$
a b c

Explanation:

Step1: Use mid - point property

Since \(B\) is the mid - point of \(\overline{AC}\), then \(AB = BC\) and \(AC=AB + BC = 2BC\). Given \(AC = 8x-20\) and \(AB=3x - 1\), and \(BC=AB\) (because \(B\) is the mid - point), also \(AC = 2AB\). So \(8x-20=2(3x - 1)\).

Step2: Solve the equation for \(x\)

Expand the right - hand 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 both sides by 2: \(x = 9\).

Step3: Find \(BC\)

Since \(BC=3x - 1\), substitute \(x = 9\) into the expression. \(BC=3\times9-1=27 - 1=26\).

Answer:

26