Question

find an equation for the perpendicular bisector of the line segment whose endpoints are (-8,-5) and (-4,1)

Answer 1

y = -x - 8

Explanation:

Midpoint of the line:

x = (x1 + x2)/2 = (-8 + -4)/2 = -12/2 = -6

y = (y1 + y2)/2 = (-5 + 1)/2 = -4/2 = -2

so midpoint is (-6,-2)

Slope of the line: Slope m = (y2-y1)/(x2-x1)

m = (-1 - -5)/(-4 - -8) = (-1 + 5)/(-4 + 8) = 4/4 = 1

Perpendicular lines have slopes that are negative reciprocals of one another

so slope of the perpendicular line is -1/1 is -1

y = mx + b

y = -x + b

Using (-6,-2)

-2 = -(-6) + b

-2 = 6 + b

b = -8

so y = -x - 8

Answer 2

`y=(3)/(2)x-3`

Explanation:

Step 1: Perpendicular bisector

To find the perpendicular bisector of the segment, apply the midpoint formula:

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

Points: {(-8, -5, (-4, 1)}

x₁ = -8 first x value

x₂ = -4 second x value

y₁ = -5 first y value

y₂ = 1 second y value

Plug the points into the formula:

`\bigg((-8+(-4))/(2), (-5+1)/(2)\bigg)`

Solve:

`\bigg((-8+(-4))/(2), (-5+1)/(2)\bigg)`

`=\bigg((-12)/(2), (-4)/(2)\bigg)`

`=(-6,-2)`

The midpoint is (-6, -2).

Step 2: Slope

To find the slope (m), apply the formula:

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

(point location is the same as previous step)

Plug the points into the formula; then solve:

`(1-(-5))/(-4(-8))`

`=(6)/(4)`

`m=(3)/(2)`

The slope is 3/2

Step 3: Solving for b

`y = mx+b`

`-6=(3)/(2)(-2)+b`

`(3)/(2)\left(-2\right)+b=-6`

`-(3)/(2)\cdot \:2+b=-6`

`-3+b=-6`

`-3+b+3=-6+3`

`b=-3`

Therefore, the equation is
`\bold{-6=(3)/(2)(-2)-3}`