Question

if $overline{ab}$ is defined by the endpoints $a(4,2)$ and $b(8,6)$, write an equation of the line that is the perpendicular bisector of $overline{ab}$.

Explanation:

Step1: Find the 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 $A(4,2)$ and $B(8,6)$, the mid - point $M$ is $(\frac{4 + 8}{2},\frac{2+6}{2})=(6,4)$.

Step2: Find the 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 $A(4,2)$ and $B(8,6)$, the slope of $AB$ is $m_{AB}=\frac{6 - 2}{8 - 4}=\frac{4}{4}=1$.

Step3: Find the slope of the perpendicular bisector

If two lines are perpendicular, the product of their slopes is $- 1$. Let the slope of the perpendicular bisector be $m_p$. Since $m_{AB}\times m_p=-1$ and $m_{AB} = 1$, then $m_p=-1$.

Step4: Write the equation of the perpendicular bisector

The point - slope form of a line is $y - y_0=m(x - x_0)$, where $(x_0,y_0)$ is a point on the line and $m$ is the slope of the line. Using the point $M(6,4)$ and $m_p=-1$, we have $y - 4=-1(x - 6)$.
Simplifying gives $y-4=-x + 6$, or $x+y-10 = 0$.

Answer:

$x + y-10=0$