Question

How do you find the equation for the perpendicular bisector of the segment with endpoints `(-1,-3)` and `(7,1)`?

Answer 1

`y=-2x+5`

Explanation:

`"the perpendicular bisector bisects the line segment at"`
`"right angles"`

`"we require to find the midpoint of the segment and "`
`"the slope m"`

`"the midpoint of any endpoints say "(x_1,y_1)" and "(x_2,y_2)" is"`

`•color(white)(x)[1/2(x_1+x_2),1/2(y_1+y_2)]`

`"midpoint "=[1/2(-1+7),1/2(-3+1)]`

`color(white)("midpoint ")=[1/2(6),1/2(-2)]=(3,-1)`

`"calculate slope m using the "color(blue)"gradient formula"`

`•color(white)(x)m=(y_2-y_1)/(x_2-x_1)`

`"let "(x_1,y_1)=-1,-3)" and "(x_2,y_2)=(7,1)`

`rArrm=(1-(-3))/(7-(-1))=4/8=1/2`

`"given a line with slope m then the slope of a line"`
`"perpendicular to it is"`

`•color(white)(x)m_(color(red)"perpendicular")=-1/m`

`rArrm_"perpendicular"=-1/(1/2)=-2`

`"the equation of a line in "color(blue)"point-slope form"` is.

`•color(white)(x)y-y_1=m(x-x_1)`

`"where m is the slope and "(x_1,y_1)" a point on the line"`

`"using "m=-2" and "(x_1,y_1)=(-1,-3)" then"`

`rArry+1=-2(x-3)larrcolor(red)"in point-slope form"`

`rArry+1=-2x+6`

`rArry=-2x+5larrcolor(red)"in slope-intercept form"`