Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are (8, -6) and (-2,4).

Answer 1

The perpendicular bisector will be a line that have a slope that is the negative reciprocal of the slope of the segment and will pass through the midpoint of the segment.

We start by calculating the slope of the segment with endpoints (8,-6) and (-2,4):

`m=(y_2-y_1)/(x_2-x_1)=(-6-4)/(8-(-2))=(-10)/(10)=-1`

The perpendicular line will have a slope that is the negative reciprocal of m=1.

We can calculate the slope of the perpendicular line as:

`m_p=-(1)/(m)=-(1)/(-1)=1`

The slope of the perpendicular line is mp=1.

Now, we calculate the midpoint of the segment.

The coordinates of the midpoint will be the average between the coordinates of the two endpoints:

`\begin{gathered} x_m=(x_1+x_2)/(2)=(8+(-2))/(2)=(6)/(2)=3 \\ y_m=(y_1+y_2)/(2)=(-6+4)/(2)=-(2)/(2)=-1 \end{gathered}`

The midpoint is (3,-1).

Then, the equation of a line with slope mp=1 and a known point (3,-1) can be written in slope-point form as:

`\begin{gathered} y-y_0=m_p(x-x_0) \\ y-(-1)=1\cdot(x-3) \\ y+1=x-3 \\ y=x-3-1 \\ y=x-4 \end{gathered}`

Answer: The equation is y=x-4