Question

the coordinates of the endpoints of $overline{ab}$ are a(0,0) and b(0,6). the equation of the perpendicular bisector of $overline{ab}$ is 1) x = 0 2) x = 3 3) y = 0 4) y = 3

Explanation:

Step1: Find mid - point of AB

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For $A(0,0)$ and $B(0,6)$, the mid - point $M$ is $(\frac{0 + 0}{2},\frac{0+6}{2})=(0,3)$.

Step2: Find slope of AB

The slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For $A(0,0)$ and $B(0,6)$, $x_1=x_2 = 0$, so the slope of $AB$ is undefined (vertical line).

Step3: Find slope of perpendicular bisector

If the slope of a line is undefined, the slope of the perpendicular line is 0.

Step4: Write equation of perpendicular bisector

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using the mid - point $(0,3)$ and $m = 0$, we have $y-3=0(x - 0)$, which simplifies to $y = 3$.

Answer:

$y = 3$