Question

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

Explanation:

Step1: Use mid - point property

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

Step2: Substitute the expressions

Substitute the expressions for \(AB\) and \(AC\) into the equation \(AC = 2AB\). We get \(8x-20 = 2(3x - 1)\).

Step3: Expand the right - hand side

Expand \(2(3x - 1)\) to get \(6x-2\). So the equation becomes \(8x-20=6x - 2\).

Step4: Solve for x

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

Step5: Find BC

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

Answer:

26