Question

if b is the mid - point of ac, and ac = 8x - 20, find bc.
a 3x - 1 b c

Explanation:

Step1: Use mid - point property

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

Step2: Set up an equation

Because \(AC = 2AB\) (by mid - point property), we substitute the expressions: \(8x-20 = 2(3x - 1)\).
Expand the right - hand side: \(8x-20=6x - 2\).

Step3: Solve for \(x\)

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

Step4: Find \(BC\)

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

Answer:

26