Question

Find an equation for the perpendicular bisector of the line segment whose endpoints

Answer 1

Answer:

y= -2x -8

Step-by-step explanation:

I will be writing the equation of the perpendicular bisector in the slope-intercept form which is y=mx +c, where m is the gradient and c is the y-intercept.

A perpendicular bisector is a line that cuts through the other line perpendicularly (at 90°) and into 2 equal parts (and thus passes through the midpoint of the line).

Let's find the gradient of the given line.

$\boxed{gradient = \frac{y1 -y 2}{x1 - x2} }$

Gradient of given line

$= \frac{1 - ( - 5)}{3 - ( - 9)}$

$= \frac{1 + 5}{3 + 9}$

$= \frac{6}{12}$

$= \frac{1}{2}$

The product of the gradients of 2 perpendicular lines is -1.

(½)(gradient of perpendicular bisector)= -1

Gradient of perpendicular bisector

= -1 ÷(½)

= -1(2)

= -2

Substitute m= -2 into the equation:

y= -2x +c

To find the value of c, we need to substitute a pair of coordinates that the line passes through into the equation. Since the perpendicular bisector passes through the midpoint of the given line, let's find the coordinates of the midpoint.

$\boxed{midpoint = ( \frac{x1 + x2}{2} , \frac{y1 + y2}{2}) }$

Midpoint of given line

$= ( \frac{3 - 9}{2} , \frac{1 - 5}{2} )$

$= ( \frac{ - 6}{2} , \frac{ - 4}{2} )$

$= ( - 3 , - 2)$

Substituting (-3, -2) into the equation:

-2= -2(-3) +c

-2= 6 +c

c= -2 -6 (-6 on both sides)

c= -8

Thus, the equation of the perpendicular bisector is y= -2x -8.