Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are (-1,-6) and (-7,2).

Answer 1

-3x+4y=4

Step-by-step explanation:

Step 1: Find the slope of the line joining points (-1,-6) and (-7,2).​

`\begin{gathered} \text{Slope}=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}} \\ =(-6-2)/(-1-(-7)) \\ =(-8)/(-1+7) \\ =-(8)/(6) \\ m=-(4)/(3) \end{gathered}`

Step 2: Find the midpoint of the line segment.

`\begin{gathered} \text{Midpoint}=\mleft((-1+(-7))/(2),(-6+2)/(2)\mright) \\ =\mleft((-1-7)/(2),(-6+2)/(2)\mright) \\ =\mleft(-(8)/(2),-(4)/(2)\mright) \\ =(-4,-2) \end{gathered}`

Step 3: Find the slope of the perpendicular line.

Two lines are perpendicular if the product of their slopes = -1.

Let the slope of the perpendicular bisector =n

`\begin{gathered} mn=-1 \\ -(4)/(3)n=-1 \\ n=(3)/(4) \end{gathered}`

Step 4: Find the equation for the perpendicular bisector.

This is the equation of the line with a slope of 3/4 passing through (-4,-2).

`\begin{gathered} y-y_1=n(x-x_1) \\ y-(-2)=(3)/(4)(x-(-4)) \\ y+2=(3)/(4)(x+4) \\ y=(3)/(4)(x+4)-2 \\ y=(3)/(4)x+3-2 \\ y=(3)/(4)x+1 \end{gathered}`

The equation for the perpendicular bisector is:

`\begin{gathered} y=(3)/(4)x+1 \\ y=(3x+4)/(4) \\ 4y=3x+4 \\ -3x+4y=4 \end{gathered}`