Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are (3, -8) and (7,2).

Answer 1

`y=-(2)/(5)x-1`

Step-by-step explanation

We want to find the equation of the perpendicular bisector of the line segment with the given endpoints.

To do this, we first have to find the slope of the given line, since the slope of a line perpendicular to a given line is the negative inverse of the slope of the line.

Then, we have to find the midpoint of the given line since the line is a bisector, it passes through the midpoint of the given line segment.

To find the slope of the line, apply the formula for the slope of a line:

`m=(y_2-y_1)/(x_2-x_1)`

where (x1, y1) and (x2, y2) are the two endpoints of the line segment

Hence, the slope of the line is:

`\begin{gathered} m=(2-(-8))/(7-3)=(2+8)/(7-3) \\ m=(10)/(4) \\ m=(5)/(2) \end{gathered}`

The negative inverse of this is:

`\begin{gathered} -((1)/((5)/(2))) \\ \Rightarrow-(2)/(5) \end{gathered}`

To find the midpoint of the endpoints, apply the formula for midpoint:

`((x_1+x_2)/(2),(y_1+y_2)/(2))`

Hence, the midpoint of the given endpoints are:

`\begin{gathered} ((3+7)/(2),(-8+2)/(2)) \\ \Rightarrow((10)/(2),(-6)/(2)) \\ \Rightarrow(5,-3) \end{gathered}`

Now, we have the slope and an endpoint of the perpendicular bisector.

To find the equation of the line, we have to apply the point-slope method:

`y-y_1=m(x-x_1)`

Therefore, the equation of the perpendicular bisector of the line segment is:

`\begin{gathered} y-(-3)=-(2)/(5)(x-5) \\ y+3=-(2)/(5)x+2 \\ y=-(2)/(5)x+2-3 \\ y=-(2)/(5)x-1 \end{gathered}`