Question

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

Answer 1

Answer:

$y=\frac{3}{2}x-3$

Step-by-step explanation:

Step 1: Perpendicular bisector

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

$\bigg(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\bigg)$

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:

$\bigg(\frac{-8+(-4)}{2}, \frac{-5+1}{2}\bigg)$

Solve:

$\bigg(\frac{-8+(-4)}{2}, \frac{-5+1}{2}\bigg)$

$=\bigg(\frac{-12}{2}, \frac{-4}{2}\bigg)$

$=(-6,-2)$

The midpoint is (-6, -2).

Step 2: Slope

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

$\frac{y_2-y_1}{x_2-x_1}$

(point location is the same as previous step)

Plug the points into the formula; then solve:

$\frac{1-(-5)}{-4(-8)}$

$=\frac{6}{4}$

$m=\frac{3}{2}$

The slope is 3/2

Step 3: Solving for b

$y = mx+b$

$-6=\frac{3}{2}(-2)+b$

$\frac{3}{2}\left(-2\right)+b=-6$

$-\frac{3}{2}\cdot \:2+b=-6$

$-3+b=-6$

$-3+b+3=-6+3$

$b=-3$

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

Answer 2

Answer:

y = -x - 8

Step-by-step 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