How do you find the equation that is the perpendicular bisector of the line segment with endpoints #(-2, 4)# and #(6, 8)#?

Question

Autumn Armstrong · Accepted Answer

Equation of the perpendicular bisector: #y = -2x+10#

The slope of the line connecting these points must first be determined:

#m = (y_2-y_1)/(x_2-x_1) = (8-4)/(6-(-2)) = 4/8 = 1/2#

The line perpendicular to this will have #m = -2#

The midpoint's coordinates are also required.

#M((x_1+x_2)/2 ; (y_1+y_2)/2)#

#M((-2+6)/2 ; (4+8)/2)#

#M(2,6)#

Use the formula for slope and one point: #y-y_1 = m(x-x_1)#

#y -6 = -2(x-2)#

#y = -2x+4+6#

Equation of the perpendicular bisector: #y = -2x+10#

Mason Adams · Answer

undefined To find the equation of the perpendicular bisector of a line segment with endpoints $(-2, 4)$ and $(6, 8)$, follow these steps:

1. Find the midpoint of the line segment using the midpoint formula: 
$$
\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)
$$
2. Determine the slope of the line segment using the slope formula: 
$$
m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
$$
3. Find the negative reciprocal of the slope calculated in step 2 to get the slope of the perpendicular bisector.
4. Use the midpoint found in step 1 and the slope from step 3 in the point-slope form of the equation of a line:
$$
y - y_1 = m(x - x_1)
$$
5. Simplify the equation obtained in step 4 to its standard form, if required.

Following these steps, you can find the equation of the perpendicular bisector of the line segment.