Question

task card 3
triangle abc has vertices a(0,0), b(6,8), and c(8,4). which equation
represents the perpendicular bisector of bc?
task card 4
determine the distance between point a(-1,-3) and point b(5,5). write an
equation of the perpendicular bisector of ab.

Explanation:

Step1: Find the mid - point of AB

The mid - point formula is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For $A(-1,-3)$ and $B(5,5)$, the mid - point $M$ is $(\frac{-1 + 5}{2},\frac{-3+5}{2})=(2,1)$.

Step2: Calculate the slope of AB

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For $A(-1,-3)$ and $B(5,5)$, $m_{AB}=\frac{5-(-3)}{5-(-1)}=\frac{8}{6}=\frac{4}{3}$.

Step3: Determine the slope of the perpendicular bisector

The slope of a line perpendicular to a line with slope $m$ is $m'=-\frac{1}{m}$. So the slope of the perpendicular bisector of $AB$ is $m =-\frac{3}{4}$.

Step4: Use the point - slope form to find the equation of the perpendicular bisector

The point - slope form is $y - y_0=m(x - x_0)$. Using the mid - point $(2,1)$ and slope $-\frac{3}{4}$, we have $y - 1=-\frac{3}{4}(x - 2)$.
Expand it: $y-1=-\frac{3}{4}x+\frac{3}{2}$.
Then $y=-\frac{3}{4}x+\frac{3}{2}+1=-\frac{3}{4}x+\frac{5}{2}$. Multiply through by 4 to get $4y=-3x + 10$, or $3x+4y=10$.

Answer:

The equation of the perpendicular bisector of $AB$ is $3x + 4y=10$.