Question

1. if b is the midpoint 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 \(AC=AB + BC = 2AB=2BC\). So \(AB=\frac{1}{2}AC\). Given \(AB = 3x - 1\) and \(AC = 8x-20\), we have \(3x - 1=\frac{1}{2}(8x - 20)\).

Step2: Solve the equation for x

First, expand the right - hand side: \(3x - 1 = 4x-10\). Then, move the terms with x to one side: \(4x-3x=10 - 1\). So \(x = 9\).

Step3: Find the length of AC

Substitute \(x = 9\) into the formula for \(AC\): \(AC=8x - 20=8\times9-20=72 - 20=52\).

Step4: Find the length of BC

Since \(BC=\frac{1}{2}AC\), then \(BC=\frac{1}{2}\times52 = 26\).

Answer:

26