Question

b is the mid - point of $overline{ac}$. $bc = 8z - 2$ and $ac = 12$. find $z$, $ab$, and $bc$.

Explanation:

Step1: Use mid - point property

Since B is the mid - point of AC, then $AB = BC$ and $AC=AB + BC = 2BC$.

Step2: Set up equation for BC

Given $AC = 12$ and $AC = 2BC$, we have $2BC=12$, so $BC=\frac{AC}{2}=\frac{12}{2}=6$.

Step3: Solve for z

Since $BC = 8z - 2$ and $BC = 6$, we set up the equation $8z-2 = 6$. Add 2 to both sides: $8z=6 + 2=8$. Then divide both sides by 8, so $z = 1$.

Step4: Find AB

Since $AB = BC$ and $BC = 6$, then $AB = 6$.

Answer:

$z = 1$
$AB = 6$
$BC = 6$