Question

find the equation in slope intercept form of a line that is a perpendicular bisector of segment AB with endpoints A(-5,5) and B(3,-3)

Answer 1

The equation in slope intercept form of a line that is a perpendicular bisector of segment AB with endpoints A(-5,5) and B(3,-3) is y = x + 2

Solution:

Given, two points are A(-5, 5) and B(3, -3)

We have to find the perpendicular bisector of segment AB.

Now, we know that perpendicular bisector passes through the midpoint of segment.

The formula for midpoint is:

$\text { midpoint }=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

$Here x_1 = -5 ; y_1 = 5 ; x_2 = 3 ; y_2 = -3$

$\text { So, midpoint of } A B=\left(\frac{-5+3}{2}, \frac{5+(-3)}{2}\right)=\left(\frac{-2}{2}, \frac{2}{2}\right)=(-1,1)$

Finding slope of AB:

$\text { Slope of } A B=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$\text { Slope } m=\frac{-3-5}{3-(-5)}=\frac{-8}{8}=-1$

We know that product of slopes of perpendicular lines = -1  

So, slope of AB $\times$ slope of perpendicular bisector = -1  

- 1 $\times$ slope of perpendicular bisector = -1  

Slope of perpendicular bisector = 1

We know its slope is 1 and it goes through the midpoint (-1, 1)

The slope intercept form is given as:

y = mx + c

where "m" is the slope of the line and "c" is the y-intercept

Plug in "m" = 1

y = x + c   ---- eqn 1

We can use the coordinates of the midpoint (-1, 1) in this equation to solve for "c" in eqn 1

1 = -1 + c

c = 2

Now substitute c = 2 in eqn 1

y = x + 2

Thus y = x + 2 is the required equation in slope intercept form